Numerical study on the dopant concentration and refractive index profile evolution in an optical fiber manufacturing process

$Page 1: Numerical study on the dopant concentration and refractive index profile evolution in an optical fiber manufacturing process$
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International Journal of Heat and Mass Transfer 49 (2006) 2097–2112

Numerical study on the dopant concentration and refractive indexprofile evolution in an optical fiber manufacturing process

Y. Yan, R. Pitchumani *

Advanced Materials and Technologies Laboratory, Department of Mechanical Engineering, University of Connecticut,

191 Auditorium Road, Unit 3139, Storrs, CT 06269-3139, USA

Received 2 October 2005; received in revised form 23 November 2005Available online 23 March 2006

Abstract

The index of refraction is an important property of optical fibers, since it directly affects the bandwidth and optical loss during infor-mation transmission. The refractive index is governed by the dopant concentration distribution across the fiber cross section, which isstrongly influenced by the processing conditions. An understanding of the effects of process parameters on the dopant concentration pro-file evolution is important to design the drawing process for tailored refractive index and optical transmission characteristics. Althoughthe heat and momentum transport in optical fiber drawing have been studied extensively, little has been reported in the open literature ondopant concentration and index of refraction profile development during processing. This paper presents a two-dimensional numericalanalysis on the flow, heat and mass transfer phenomena involved in the drawing and cooling process of glass optical fibers using a finitedifference approach based on primitive variables. The effects of several important parameters are investigated in terms of nondimensionalgroups, including: fiber draw speed, inert gas velocity, furnace dimensions, gas properties, and dopant properties on the flow, temper-ature and dopant concentration distribution.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Optical fiber; Fiber drawing; Fiber cooling; Refractive index; Dopant diffusion

1. Introduction

Optical fibers, both multimode and single mode, arecritical media for transmitting light signals used as infor-mation carriers in the communications industry. Formultimode fibers, the two basic parameters governing thetransmission behavior of the waveguides are fiber corediameter and core refractive index profile. The refractiveindex profile must be carefully controlled in order toachieve its optimal information carrying capacity [1]. Forsingle mode fibers, cladding diameter is also an importantparameter, and a minimum value must be maintained toavoid excessive losses [2]. The refractive index of opticalfiber is modified by doping pure silica (SiO2) with materials

0017-9310/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2005.11.030

* Corresponding author. Tel.: +1 860 486 0683; fax: +1 860 486 5088.E-mail address: [email protected] (R. Pitchumani).

such as oxides of germanium, phosphorous and boron.Germanium (Ge) and phosphorous (P) increase the refrac-tive index of SiO2, while boron (B) decreases it. Since afiber can guide light only if the refractive index of the coreis higher than that of the surrounding region, most fibersconsist of a cladding region of pure SiO2 and a core dopedwith GeO2 or P2O5 to increase its refractive index, while insome fibers the core is pure SiO2 and the cladding is dopedwith B2O3 to decrease its refractive index.

The manufacturing process for optical fibers involvespreform fabrication, optical fiber drawing, cooling andcoating. The preform is preferentially made by the so-called modified chemical vapor deposition process, whichincludes the deposition of dopants within a fused quartzglass tube by oxidation of gaseous halides at temperaturesof 1700–2100 K and subsequent collapsing of the tube intoa rod [3]. The preform is then heated to above its meltingtemperature, while an axial tension is simultaneously

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Nomenclature

c dopant concentration expressed as a mass frac-tion

cp specific heat (J/kg K)D diffusion coefficient (m2/s)h heat transfer coefficient for natural convective

cooling (W/m2 K)k thermal conductivity (W/m K)kB Boltzmann constant (1.3807 · 1023 J/K)l length of the drawing and cooling regions (m)n refractive indexp pressure (N/m2)Pe Peclet number for dopant diffusionPr Prandtl numberr radial coordinate, radius (m)Re Reynolds numberT physical temperature in absolute scale (K)u axial velocity component (m/s)v radial velocity component (m/s)z axial coordinate (m)

Greek symbols

b dimensionless axial coordinate in drawing andcooling region

g dimensionless radial coordinatel dynamic viscosity (N s/m2)m kinematic viscosity (m2/s)q density (kg/m3)h dimensionless temperature

Subscriptsa inert gasc cooling gasdr drawing regionf fiberF furnacefc forced convection cooling regionm melting point of fibernc natural convection cooling regionw wall temperature of the cooling channel

2098 Y. Yan, R. Pitchumani / International Journal of Heat and Mass Transfer 49 (2006) 2097–2112

applied to draw the preform into a fiber, a process calleddrawing. The heating is usually accomplished by feedingthe preform into a concentric, cylindrical electric furnacewhere radiation is the dominant mode of heat transfer[4]. It is during this process that the preform/fiber encoun-ters a drastic change in diameter, temperature, and dopantconcentration distribution. Upon exiting the furnace, thefiber undergoes a cooling process, where its temperatureis lowered from around its melting temperature of approx-imately 1900 K to below 500 K [5] through convection andradiation. If the drawing speed of the fiber is very high,forced convective cooling is necessary since natural convec-tion alone will be insufficient to reduce the fiber tempera-ture to the level required by the subsequent coatingprocess. In the coating process, a jacket of polymer mate-rial is added to encase the fiber and protect it fromscratches that would weaken the fiber’s strength. Sincethe diffusivity of the dopant is very small at low tempera-tures, the dopant distribution is essentially stabilized atthe end of the cooling section and remains unchanged inthe coating process. Therefore, the main focus of this studyis on the transport phenomena in the optical fiber drawingand cooling regions of the manufacturing process.

In the last three decades, there have been several studieson the temperature profile evolution during the drawingand cooling steps [4–21]. Paek [17] numerically simulatedthe drawing and cooling process of optical fibers assuminga lumped temperature distribution along the radius, a pre-scribed temperature for the fiber at the furnace exit, and aprescribed heat transfer coefficient between the fiber and itsenvironment. Jaluria and coworkers [5–9,15,20] have car-ried out more rigorous studies on the neck-down regionby solving the coupled convection, conduction and radia-

tion transport between the furnace muffle and the neckedpreform. Their results provide detailed information onthe heat transfer mechanism in fiber drawing, but in thesestudies the momentum equations were solved using thestream function–vorticity approach, which is limited totwo-dimensional analysis.

Glicksman [11] performed analytical and experimentalstudies on the cooling process of optical fibers. In hismodel, a lumped capacity method was used to predict thetransient temperature variation in the fiber, while the Nus-selt number for the natural convective cooling process wassimulated using the Reynolds analogy. Choudhury et al. [5]carried out numerical simulation on the forced cooling ofoptical fibers prescribing a uniform initial temperature dis-tribution at the entrance of the cooling section. The com-plete two-dimensional momentum and energy equationsfor the fiber as well as the cooling gas were solved usinga finite difference method based on stream function–vortic-ity approach. Gossiaux et al. [12] studied forced cooling ofoptical fibers using a nitrogen–helium mixture as the cool-ing gas. The effects of the cooling tube diameter and cool-ing gas property on the cooling rate were studied.

Scarce information is available in the open literature onthe theoretical, computational, or experimental study ofdopant diffusion during the optical fiber manufacturingprocess and, consequently, the refractive index prediction.Palmer [22] analytically studied the evolution of tempera-ture and dopant distribution in the cooling of optical fibersassuming a heat transfer coefficient between the fiber andthe forced cooling gas.

These studies have provided much information on theheat transfer phenomena during the fiber manufacturingprocess, but the models have generally ignored the com-

r v Furnacezu

uf

hh

heating zone

natural cooling

forced cooling

Fig. 1. Schematic of an optical fiber drawing process along the furnaceand the cooling zone.

Y. Yan, R. Pitchumani / International Journal of Heat and Mass Transfer 49 (2006) 2097–2112 2099

bined effects of the drawing and cooling processes, insteadrelying on an assumed temperature variation at thejunction of the drawing and cooling sections. In order toprovide a more realistic solution to the temperature evolu-tion in preform and fiber, this research investigates theconjugate heat transfer problem during the drawing andcooling processes. This study also seeks to address the needfor rigorous investigation of dopant diffusion during themanufacturing process of optical fibers, which is presentlylacking in the literature.

The objective of this study is to develop a mathematicalmodel for the prediction of the temperature and dopantconcentration evolution during the drawing and coolingprocesses, and the refractive index profile in the fiber. Tothis end, the model describes the convection, conduction,radiation and species transport during the combined draw-ing and cooling process. Allied with this objective is thedevelopment of a computational framework, which solvesthe mathematical model formulated in terms of primitivevariables, using a control volume based finite differencescheme. Parametric studies are presented to illustrate theeffects of the various dimensionless parameters on the tem-perature and dopant concentration evolution, and therefractive index profile in the paper. Section 2 describesthe formulation of the governing equations and the numer-ical solution procedure. The results of the study are pre-sented and discussed in Section 3.

2. Analysis

Fig. 1 shows a schematic of the drawing and cooling sec-tions considered for the modeling. In the drawing process,laminar axisymmetric flows of the glass preform/fiber andinert gas in a cylindrical furnace are studied. For simplicity,it is assumed that the flow and temperature fields are notsignificantly affected by the dopant, which is taken asGeO2 in this study.

Considering the process to be at steady state, the gov-erning equations for mass, momentum, energy and dopanttransport can be expressed as

Continuity equation:

oðquÞozþ 1

roðqrvÞ

or¼ 0 ð1Þ

Momentum equations:

o qu2ð Þoz

þ 1

roðqruvÞ

or

¼ � opozþ 2

o

ozl

ouoz

� �þ 1

ro

orrl

ouorþ ov

oz

� �� ð2Þ

oðquvÞoz

þ 1

roðqrv2Þ

or

¼ � oporþ o

ozl

ouorþ ov

oz

� �� þ 2

ro

orrl

ovor

� �� 2l � v

r2ð3Þ

Energy equation:

oðqcpuT Þoz

þ 1

roðqcprvT Þ

or

¼ o

ozkoToz

� �þ 1

ro

orrk

oTor

� �

þ l 2ouor

� �2

þ ur

� �2

þ ovoz

� �2" #(

þ ouozþ ov

or

� �2)

ð4Þ

Dopant diffusion equation:

oðucÞozþ 1

roðrvcÞ

or¼ o

ozD

ocoz

� �þ 1

ro

orrD

ocor

� �ð5Þ

where all the terms are defined in the nomenclature. Theabove equations are applicable for the entire drawing andcooling sections, with the exception of the energy equation(Eq. (4)), in which viscous dissipation is negligible in theinert gas in the furnace and cooling sections.

2.1. Drawing region

In the furnace, the diameter of the preform changesdrastically in the axial direction. The glass preform has astreamwise variation in radius r(z) as it undergoes necking


into a fiber. This profile depends on the forces acting at thesurface and should be obtained as a part of the solution.However, in the interest of reducing the computationalintensity of the simulations, a known necking shapereported in the literature is assumed. Both natural convec-tive cooling and forced convective cooling are studied; fornatural convective cooling, a known convective coefficientis assumed here using experimental results cited in existingliterature, whereas for forced convective cooling, the com-plete governing equations are solved for fiber and coolinggas. In this study, the neck-down profile is chosen to beconsistent with available data from the literature. Thelengths of the furnace and the preform are both taken tobe 0.3 m, the necking occurs between the distance of0.04 m and 0.22 m. The necking shape of the preform/fiberis described as the logarithmic function of an eighth-orderpolynomial [15].

For z6 0:04 m: rðzÞ¼ 0:006288 m

For z P 0:22m: rðzÞ¼ 0:000059 m

For 0:04 m6 z6 0:22 m:

log10rðzÞ¼�6:66531�106ðz�0:02Þ8þ1:5152�107ðz�0:02Þ7

�6:20906�106ðz�0:02Þ6þ1:56512�105ðz�0:02Þ5

þ3:0826�105ðz�0:02Þ4�4:71803�104ðz�0:02Þ3

þ7:884�102ðz�0:02Þ2þ15:7224ðz�0:02Þ�22:31642

ð6ÞThe complex physical domain is transformed into a rightcylindrical domain amenable to a computational imple-mentation using the Laudau’s transformation applied tothe preform/fiber and inert gas within the furnace.

For preform/fiber:

b ¼ zldr

and g ¼ rrðzÞ ð7Þ

For inert gas:

b ¼ zldr

and g ¼ rF � rra

¼ f ðkr; raÞ ð8Þ

where

kr ¼ rF=ra and ra ¼ rF � rðzÞOne side effect of the coordinate transformation as de-scribed in the above paragraphs is that the mass, momen-tum and energy conservation are not preserved whencontrol volume based finite difference method is appliedto solve the generated governing equations, since the veloc-ity field is not orthogonal to the computational domain.This problem can be resolved by the use of curvilinearvelocities, which preserves mass, momentum, and energyconservation in the total flow field [23–26]. Therefore, thefollowing velocity transformations are employed:

For preform/fiber:

�u ¼ u and �v ¼ �gdrðzÞ

dzuþ v ð9Þ

For inert gas:

�u ¼ u and �v ¼ gdrðzÞ

dzu� v ð10Þ

In order to describe the flow, energy, and mass transport ina more generic way, the governing equations are also non-dimensionalized by using the following nondimensionalgroups:

For preform/fiber:

U ¼ �uuf

; V ¼ �vuf

; P ¼ pqf ;mu2

f

; h ¼ TT m

; C ¼ cc0

ð11Þ

Ref ¼uf rf

mf ;m

; Prf ¼mf ;m

af ;m

; Pe ¼ ufrf

Dm

; RðzÞ ¼ rðzÞrf

ð12Þ

For inert gas:

U ¼ �uua;0

; V ¼ �vua;0

; P ¼ pqa;mu2

a;0

; h ¼ TT m

ð13Þ

Rea ¼ua;0rF

ma;m

; Pra ¼ma;m

aa;m

; RF ¼rF

rf

ð14Þ

where ua,0 is the inert gas velocity at the entrance of thefurnace.

Thermophysical properties are nondimensionalizedusing those at the reference temperature, the melting tem-perature of the fiber

fq ¼qqm

; f Cp¼ Cp

Cp;m; f k ¼

kkm

; f m ¼mmm

; f D ¼D

Dm

ð15Þ

The nondimensionalized governing equations in the pre-form/fiber can be expressed as


1

Ldr

obR2ðzÞgUcob

þ o½RðzÞgV �og

¼ 0 ð16Þ

Momentum equations:

1

Ldr

o RðzÞ2gU 2j k

obþ o½RðzÞgUV �

og

¼ �RðzÞ2gLdr

oPobþ RðzÞg2

Ldr

dRðzÞdb

oPog

þ 2RðzÞ2gRef L2

dr

o

obfm

oUob

� �þ 1

Ref

1þ 2g2

L2dr

dRðzÞdb

� �2( )

� o

ogfmg

oUog

� �� 1

Ref L2dr

dRðzÞdb

� �2o

ogfmg

3 oUog

� �

� 2RðzÞ2g2

Ref L2dr

o

obfm

RðzÞdRðzÞ

dboUog

� �

� 2RðzÞg2

Ref L2dr

dRðzÞdb

o

ogfm

oUob

� �þ RðzÞ

Ref L2dr

dRðzÞdb

o

ogfmg

2 oUob

� �


þ 1

Ref L2dr

RðzÞ d2RðzÞdb2

� dRðzÞdb

� �2( )

o

ogfmg

2U�

þ RðzÞRef Ldr

o

ogfmg

oVob

� �� 1

Ref Ldr

dRðzÞdb

o

ogfmg

2 oVog

� �ð17Þ

1

Ldr

o RðzÞ2gUVh i

obþ

o RðzÞgV 2 �

ogþ RðzÞ2g2

L2dr

o

obdRðzÞ

dbU 2

� �

þ 2RðzÞg2

L2dr

dRðzÞdb

� �2

U 2 þ RðzÞg2

Ldr

dRðzÞdb

o

ogðUV Þ

þ 2RðzÞgLdr

dRðzÞdb

UV

¼ �RðzÞg oPogþ RðzÞ2g

Ref L2dr

o

obfm

oVob

� �

þ 1

Ref

2þ g2

L2dr

dRðzÞdb

� �2( )

o

ogfmg

oVog

� �

� RðzÞ2g2

Ref L2dr

o

obfm

RðzÞdRðzÞ

dboVog

� �� RðzÞg2

Ref L2dr

dRðzÞdb

o

ogfm

oVob

� �

þ RðzÞ2g2

Ref L3dr

o

obfm

dRðzÞdb

oUob

� �þ RðzÞ2g

Ref Ldr

o

obfm

RðzÞoUog

� �

� RðzÞ2g3

Ref L3dr

o

obfm

RðzÞdRðzÞ

db

� �2oUog

( )

þ RðzÞ2g2

Ref L3dr

o

obfm

d2RðzÞdb2

U� �

� RðzÞ2g2

Ref L3dr

o

obfm

RðzÞdRðzÞ

db

� �2

U

( )

þ 1

Ref Ldr

2þ g2

L2dr

dRðzÞdb

� �2( )

dRðzÞdb

o

ogfmg

2 oUog

� �

� g2

Ref Ldr

dRðzÞdb

o

ogfm

oUog

� �� RðzÞg2

Ref L3dr

dRðzÞdb

� �2o

ogfmg

oUob

� �

þ 1

Ref Ldr

2þ g2

L2dr

dRðzÞdb

� �2( )

dRðzÞdb

o

ogfmgUð Þ

� RðzÞg2

Ref L3dr

dRðzÞdb

d2RðzÞdb2

o

ogðfmgUÞ � 2f m

Ref Ldr

dRðzÞdb

U � 2f m

RefgV

ð18Þ

Energy equation:

1

Ldr

obRðzÞ2gUf qfCphcob

þobRðzÞgVf qfCphc

og

¼ RðzÞ2gRef PrfL2

dr

o

obfk

ohob

� �þ 1

RefPrf

1þ g2

L2dr

dRðzÞdb

� �2 !

� o

ogfkg

ohog

� �� RðzÞ2g2

RefPrf L2dr

o

obfk

RðzÞdRðzÞ

dbohog

� �

� RðzÞg2

Ref PrfL2dr

dRðzÞdb

o

ogfk

ohob

� �

þ 2f qfmEvt

Ref

goUog

� �2

þ U 2

gþ g

RðzÞgL2

dr

o

obdRðzÞ

dbU

� ��(

� g

L2dr

dRðzÞdb

� �2o

ogðgUÞ þ RðzÞ

Ldr

oVob� g

Ldr

dRðzÞdb

oVog

)2)

þ gfqfmEvt

Ref

RðzÞLdr

oUobþ 1

Ldr

dRðzÞLdrdb

U þ oVog

� �2

ð19Þ


1

Ldr

o½RðzÞ2gUC�ob

þ o½RðzÞgVC�og

¼ RðzÞ2gPeL2

dr

o

obfD

oCob

� �

þ 1

Pe1þ g2

L2dr

dRðzÞdb

� �2( )

o

ogfDg

oCog

� �

� RðzÞ2g2

PeL2dr

o

obfD

RðzÞdRðzÞ

dboCog

� �

� RðzÞg2

PeL2dr

dRðzÞdb

o

ogfD

oCob

� �ð20Þ

This set of equations applies to the preform and fiber in thedrawing region. The equations for inert gas can be easilyderived from Eqs. (16) to (19) by replacing g with kr�g

kr�1.

Clearly, the dopant diffusion equation does not apply to in-ert gas, and the nondimensionalized form of the other gov-erning equations for the inert gas can be derived from thenondimensionalized governing equations for preform/fiberby simple mathematical manipulation.

At the entrance to the furnace, the temperatures of thepreform and the inert gas are taken as constant at roomtemperature. The axial velocities of the preform/fiber andinert gas are uniform, while the radial velocities are zero.The dopant concentration is expressed as mass fractionof the dopant, and is assumed to follow a step function as

c ¼c0 for 0 < g 6 0:1

0 for 0:1 < g 6 1

�ð21Þ

where the initial dopant concentration in the preform corec0 is set to 0.2 in this study.

The velocity normal to the interface of the preform/fiberand the inert gas is zero, while the velocity along the inter-face is continuous on both sides. Velocity is obtained byapplying the continuity of shear stress at the preform/fiber–inert gas interface. The temperature is continuousand the heat flux satisfies the following condition:

koTon

� �fiber

¼ koTon

� �gas

þ qrad ð22Þ

where the radiation heat flux at the surface of the preform/fiber, qrad, is computed using the same approach proposedby Lee and Jaluria [15]. The radiation heat transfer is stud-ied using an enclosure model proposed by Lee and Jaluria[15], with the enclosure composed of furnace wall, outersurface of preform/fiber, and the inlet and outlet of theinert gas. The gas within the radiative enclosure is treatedas nonparticipating. The furnace is taken as a diffuse gray


surface with a constant diffusivity of 0.75 [16]. The inletand outlet of the inert gas are treated as black surfaces atan ambient temperature of 300 K. The silica preform isassumed to be a diffuse and spectral surface, and its hemi-spherical, spectral emissivity is represented by a fifth-orderpolynomial [15]:

e ¼ 1:73734� 10�3ð2arÞ5 � 2:7� 10�2ð2arÞ3

�0:545ð2arÞ2 þ 0:999788ð2arÞ�4:3003� 10�3 for 2ar 6 5

e ¼ 0:8852646 for 2ar P 5

where the absorption coefficient, a, is obtained using theband approximation [4] assuming that outside of the bandbetween 3 and 8 lm, the fiber is perfectly transparent. Theabsorption coefficient, a, for different wavelength, k, is asfollows: for k < 3 lm and k > 8 lm, a = 0; for 3.0 lm <k < 4.8 lm, a = 400 m�1; and for 4.8 lm < k < 8.0 lm,a = 15,000 m�1.

The radiative transport within the preform/fiber is takeninto account using diffusion approximation. Under thisapproximation, the apparent thermal conductivity of thepreform/fiber is the sum of the nonradiative conductivity,kcond, and radiative conductivity, krad, expressed as

krad ¼16n2rT 3

3að23Þ

where n is the refractive index of silica glass, r the Stefan–Boltzmann constant, and a the absorption coefficient ofsilica glass.

Zero flux condition is applied to all the transport vari-ables at the centerline of the preform/fiber. A fully devel-oped condition is used for the flow and temperature forthe inert gas at the outlet of the furnace.

2.2. Cooling region

When exiting the furnace, the fiber is near its meltingtemperature, and has to be cooled before coating can beapplied. As seen in Fig. 1, the fiber is first cooled by naturalconvection to the surrounding air. However, in high speeddrawing, natural convection alone is usually insufficient tolower the fiber temperature to the level required by thecoating process, and supplemental cooling using forcedconvection is necessary. The governing equations in thecooling region are much simpler since the speed and radiusof the fiber are constant; as a result, no coordinate or veloc-ity transformation is needed in this region. The nondimen-sionalized governing equations for energy transport anddopant diffusion in the fiber are

Energy equation:

1

Lohob¼ 1

Ref Prf

1

L2

o

obohob

� �þ 1

go

oggohog

� �� ð24Þ


1

LoCob¼ 1

Pec

1

L2

o

oboCob

� �þ 1

go

oggoCog

� �� ð25Þ

Note that since the fiber is moving with a constant, known,velocity and further since there is no fluid motion withinthe fiber, continuity and momentum equations are notneeded to describe the fiber domain. The governing equa-tions for the surrounding cooling gas include the continu-ity, momentum, and energy equations as presented belowin their nondimensional form


Rc

LoUobþ 1

goðgV Þ

og¼ 0 ð26Þ

Momentum equations:

Rc

LoðfqU 2Þ

obþ 1

goðfqgUV Þ

og

¼ �Rc

LoPobþ 1

Rec

Rc

L

� �2o

obfqfm

oUob

� �"

þ 1

go

ogfqfmg

oUog

� �#ð27Þ

Rc

LoðfqUV Þ

obþ 1

goðfqgV 2Þ

og

¼ � oPogþ 1

Rec

Rc

L

� �2o

obfqfm

oVob

� �"

þ 1

go

ogfqfmg

oVog

� �#� fqfmV

Recg2ð28Þ

Energy equation:

Rc

L

oðfqfCp UhÞoX

þ 1

g

oðfqfCpgV hÞog

¼ 1

Rec Prc

Rc

L

� �2o

obfk

ohob

� �þ 1

go

ogfkg

ohog

� �" #ð29Þ

A prescribed convection coefficient in the range of 150–300 W/m2 K [17] is applied to the fiber surface at the natu-ral convective cooling region and the boundary conditionsat the fiber/gas interface is given by

ohog¼ Biðhinf � hÞ; Bi ¼ hrf

kf

ð30Þ

where Bi is the Biot number for natural convection asdefined above.

For forced convective cooling, the gas enters the coolingtube with a uniform axial velocity at room temperature. Ano slip condition is applied to the cooling gas at the wall,where constant room temperature is assumed. At the inter-face of the fiber and the cooling gas, a continuity conditionis applied to the velocity and shear rate, temperature andheat flux, and a no-penetration condition is applied tothe dopant concentration. At the outlet of the coolingregion, the velocity, temperature and dopant concentrationprofiles are considered to be fully developed such that theiraxial gradients are zero. At the junction of the drawingregion and cooling region of the fiber, a continuity


condition is applied to temperature, dopant concentration,heat flux and mass flux. The heat flux leaving the outletinterface of the drawing region through conduction equalsthe heat flux entering the inlet interface of the coolingregion through conduction, and the dopant diffusion atthe outlet interface of the drawing region equals the dopantdiffusion at the inlet interface of the cooling region.

All the thermophysical properties of the preform/fiber,the inert gas, and the cooling gas are taken to be tempera-ture dependent. In particular, for GeO2 doped glass opticalfiber, a few different diffusion coefficient equations areavailable in the literature. Among them the most widelyused equation is the Arrhenius relation, which can beexpressed as [27–29].

D ¼ D0e�E=kBT ð31Þwhere D0 is the inert diffusion constant, E is the activationenergy, and kB is the Boltzmann constant. The values of D0

and E are obtained from the literature as D0 = 1.875 · 104

m2/s and E = 7.69 · 104 J.The refractive index of the fiber can be expressed as a

function of the dopant concentration using the Lorentz–Lorentz equation [30].

n ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2/1� /

sand / ¼

P ðn2i �1Þci

ðn2i þ2ÞqiP ciqi

ð32Þ

where ni, ci and qi are the refractive index, mass fractionand density of component i, respectively. The refractiveindices for pure SiO2 and GeO2 are 1.458 and 1.603, respec-tively [30].

The conservation equations of mass, momentum,energy, and species are coupled in the drawing regionand cooling region since the properties of the preform/fiberand inert gas are all temperature dependent. Dopant diffu-sion introduces another layer of coupling since the proper-ties of preform/fiber depend on the dopant concentration.For the purposes of this study, it is assumed that the effectsof dopant concentration on the properties of preform/fiberare small, allowing for a decoupling of the energy andmomentum equations from the dopant diffusion equation.

A control volume based finite difference method is usedto solve the governing equations and boundary conditions.A nonuniform grid mesh was used in the entire computa-tional domain, and a staggered grid structure was used in

Table 1Typical mesh grids for (a) velocity, temperature, and (b) dopant diffusion field

(a) For velocity and temperature fields Drawing region

Preform/fiber Inert ga

Number of grids in axial direction 102 102Number of grids in radial direction 42 62

(b) For dopant diffusion field Drawing region

Number of grids in axial direction 102Number of grids in radial direction 102

solving the Navier–Stokes equations to avoid unrealisticvelocity solutions. The power law scheme [31] was usedto represent the transport properties on the control volumesurfaces, and a SIMPLER algorithm was adopted to solvethe Navier–Stokes equations [31]. The coupled flow andtemperature field in the drawing and the cooling regionswere solved for using the following procedure: (a) an initialtemperature and velocity fields were assumed for the pre-form/fiber, inert gas and cooling gas in the drawing andcooling regions; (b) the temperature and velocity in the pre-form/fiber and inert gas in the drawing region were solvedfor using the governing equations; (c) the temperature andvelocity at the interface of the preform/fiber and inert gaswere determined using the boundary conditions; (d) thetemperature of the fiber was solved for in the natural con-vective cooling region; (e) the temperature of the fiber andcooling gas, and the velocity of the cooling gas were solvedfor in forced convective cooling region; (f) the temperatureat the interface of the fiber and cooling gas was solved forin forced convective cooling region; (g) the temperature atthe junctions of drawing and natural convective cooling,and natural and forced convective cooling region wasdetermined; (h) the temperature and velocity fields for allthe regions were updated; and (i) steps (b)–(h) wererepeated until convergence.

The temperature and velocity field obtained were usedto solve for the dopant concentration. The governing equa-tions in the drawing and the cooling regions differ signifi-cantly in terms of complexity due to the large variationin radius in the drawing region, therefore the computationis carried out in two separate regions simultaneously, and iscoupled as follows: (a) an initial dopant concentration isassumed at the interface of drawing and cooling regions;(b) the dopant concentration is solved for in the drawingregion; (c) the dopant concentration is obtained in the cool-ing region; (d) the dopant concentration at the interface ofdrawing and cooling regions is updated using the continu-ity conditions in dopant concentration and mass flux; and(e) steps (b)–(d) are repeated until convergence. The con-verged solution is used in Eq. (32) to get the refractiveindex profile in the fiber. In all the calculations, the conver-gence criterion of the numerical solution for the tempera-ture, velocity and dopant concentration fields was thatthe relative errors between two consecutive iterations areless than 10�7.

s

Natural cooling region Forced cooling region

s Fiber Fiber Cooling gas

42 82 8218 18 22

Natural cooling and forced cooling regions

20236

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

Current ModelChoudhury and Jaluria [8]

Sur

face

Tem

pera

ture

, θ

Dimensionless Length, β (a)

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350 400

298K, Current Model868K, Current Model298K, Choudhury et al. [5]868K, Choudhury et al. [5]

Dimensionless Length, L(b)

Sur

face

Tem

pera

ture

, θ

Fig. 2. (a) A comparison of the surface temperature profile in the furnaceregion predicted by the present model and available numerical data for anoptical fiber drawing process, and (b) a comparison of the surfacetemperature decay predicted by the present model and available numericaldata for an optical fiber cooling process.


To best capture the evolution of the temperature, thevelocity, and the dopant concentration fields while mini-mizing computational effort, different grid schemes wereused in discretizing these fields. Linear interpolation wasused to obtain the temperature and velocity values underthe different grid scheme in solving the dopant concentra-tion equations. For the parametric study described in thefollowing section, a typical number of mesh grids is shownin Table 1. For the cases studied, the total computationtime for a typical case was about 20 h on a SPARCstation

10 UNIX machine with a single processor.

3. Results and discussion

The formulation in the previous section described thegoverning equations and boundary conditions representingthe mass, momentum and energy transport in the preform/fiber, inert gas and cooling gas during the drawing andcooling steps of an optical fiber drawing process. The over-all goal of the analysis is to predict the dopant concentra-tion distribution and refractive index in the optical fiber.The computational results obtained from the analysis arepresented in this section. The model is first verified by com-parison with existing data in the open literature. With thevalidated model as basis, parametric studies are reportedto elucidate the effects of several nondimensional parame-ters on the thermal and mass transport during theprocesses.

Since no numerical or experimental results are availablein the open literature on dopant diffusion in the drawingand cooling regions or on conjugate temperature field inthe combined drawing and cooling regions, the validationis limited only to the temperature field for individual draw-ing and cooling processes. For the drawing region, the sur-face temperature is compared with numerical data fromChoudhury and Jaluria [8]. The result is shown inFig. 2a, in which the neck-down shape of the fiber is thesame as that described in the previous section, the radiusand surface temperature of the furnace are 1.9 cm and2500 K, respectively, and the inert gas is air with an inletvelocity of 5.2824 cm/s. Fig. 2a shows that in the upperneck-down region, the temperature increases rapidly fromthe ambient temperature to near the furnace temperature,and remains essentially constant through the neck-downregion. Overall, the current model prediction is seen tocompare well with the simulation results of Choudhuryand Jaluria [8] throughout the domain.

For the cooling region, the present simulations werecompared with results obtained by Choudhury et al. [5]for a glass fiber cooled by helium, and the result is shownin Fig. 2b. In this process, the diameter of the fiber is150 lm, and the speed of the fiber is 1.67 m/s. The initialtemperature of the fiber at the entrance is uniform, andthe length and width of the cooling channel are 5 and600 times of the fiber radius, respectively. The velocity ofthe cooling gas, helium, is twice the fiber speed at theentrance. In the figure, the temperature is defined as

h ¼ T�T1T 0�T1

, where the fiber temperature at the entrance,T0,is set to be 868 K, and the ambient temperature, T1 isset to be 300 K. The dimensionless distance is defined asL ¼ z

rf. Here comparisons were made for constant proper-

ties taken at two different temperatures, 298 K and868 K, respectively. Fig. 2a and b show that the results ofthe current model are in close agreement with those avail-able in the open literature.

The validated simulation model was used to investigatethe effects of several nondimensional parameters on themass, momentum and thermal transport during the draw-ing and cooling of optical fibers in order to determine thedominant parameters and their physical effects on the pro-cess. The ranges of nondimensional groups used are basedon typical practical situations for fiber drawing and cool-ing. The results on temperature and dopant distributionsare presented and discussed; however velocity fields areomitted here in the interest of brevity. For all the resultsshown, unless otherwise specified, the following conditionsare employed: The properties of the fiber and gas are taken

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

Sur

face

Tem

pera

ture

, θ

Dimensionless Length, β(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Dop

ant C

once

ntra

tion,

C

(b) Dimensionless Length, β

1.455

1.460

1.465

1.470

1.475

0.0 0.2 0.4 0.6 0.8 1.0

Ref = 4 x 10-8

Ref = 1 x 10-7

Ref = 2 x 10-7

Ref

ract

ive

Inde

x, n

Dimensionless Radius, η(c)

Fig. 3. Effects of fiber Reynolds number on (a) preform/fiber surfacetemperature, (b) preform/fiber centerline dopant concentration decay, and(c) fiber refractive index profile.


to be temperature dependent. The reference temperaturefor fiber and inert gas is the fiber melting temperature,1900 K, while the reference temperature for cooling gas istaken as ambient temperature, 300 K. The fiber velocityis 3 m/s, and the inert gas velocity at entrance is 200 timesthat of the preform velocity. The entrance temperature forpreform, inert gas and cooling gas is 300 K. The width ofthe cooling channel is 0.8 cm, and the lengths of the naturaland forced convective cooling section are 0.5 m and 1 m,respectively. The convective coefficient is taken as 300W/m2 K. The inert gas is taken as air, and the coolinggas is taken as helium. The nondimensional groups usedin the parametric study presented in Figs. 3–11 are asfollow: Ref = 1 · 10�7, Rea = 300, Rec = 500, RF = 400,hF = 1.2105, Bi = 0.0053, Rc = 200, Kc,f = Kc/Kf = 0.04,and Pe = 7.28 · 109, unless otherwise noted in the figures.

3.1. Effects of fiber Reynolds number

The effect of fiber Reynolds number on the temperaturefield, dopant concentration distribution, and refractiveindex profile is examined first. Three different fiber Rey-nolds numbers, 4 · 10�8, 1 · 10�7, and 2 · 10�7, are con-sidered. Fig. 3 shows the surface temperature (Fig. 3a)and centerline dopant concentration (Fig. 3b) as a functionof dimensionless length, b, and the refractive index profileas a function of dimensionless radius, g, in Fig. 3c. The sur-face temperature decays exponentially in the natural con-vective cooling region, as shown in Fig. 3a. In the forcedconvective cooling region, the surface temperature dropsrapidly near the entrance because of a high convective coef-ficient and large temperature difference between the fiberand the cooling gas. The surface temperature decay slowsdown along the cooling channel because of the decreasingtemperature difference between the fiber and cooling gas.Fig. 3b shows that the dopant concentration at the center-line of the fiber remains essentially constant at the upperneck-down region. This is due to the fact that the diffusioncoefficient is very small at low temperatures. The centerlinedopant concentration starts decreasing when the tempera-ture increases.

As the fiber Reynolds number increases, the fiberremains in the furnace for a shorter time and, therefore,the temperature rise is slower and smaller, as evident fromFig. 3a. Similarly, a shorter residence time in the coolingregion results in a slower surface temperature decay. Itfollows that when the fiber Reynolds number exceeds acertain value, the fiber temperature will not be able to riseto above its melting point and therefore, the drawing pro-cess will not be able to take place. On the other hand, moreeffective cooling techniques have to be resorted to at highfiber Reynolds numbers. Fig. 3b shows that dopant diffu-sion decreases with fiber Reynolds number. This is due tothe decreasing dopant diffusion coefficient as a result ofthe decreasing temperature, and the shorter time periodfor diffusion. Fig. 3c shows the refractive index profile inthe final fiber, and illustrates two important properties:

the maximum refractive index difference between the coreand cladding increases with fiber Reynolds number, andconsequently, the core radius decreases with fiber Reynoldsnumber.

3.2. Effects of inert gas Reynolds number

Fig. 4 represents the effect of inert gas Reynoldsnumber on the surface temperature, centerline dopant

Fig. 4. Effects of inert gas Reynolds number on (a) preform/fiber surfacetemperature, (b) preform/fiber centerline dopant concentration decay, and(c) fiber refractive index profile.


concentration, and refractive index profile of the fiber.Three different inert gas Reynolds numbers, 60, 300 and600, are examined. For the cases studied, as the Reynoldsnumber of the inert gas increases, fiber surface temperaturein the drawing region decreases, as seen in Fig. 4a. How-ever, as the inert gas Reynolds number gets smaller, thiseffect gets smaller and the surface temperature is hardlyaffected by the inert gas Reynolds number. When air isused as the inert gas, the surface temperature of the fiber

is higher than that of the adjacent inert gas, and therefore,the inert gas absorbs heat from the fiber. The amount ofheat absorbed by the inert gas increases with the inertgas Reynolds number due to the increased convection,and consequently, the surface temperature of the fiberdecreases with the inert gas Reynolds number. However,since the dominant mode of heat transfer in the furnaceis the radiation between the fiber and furnace wall, theeffect of convection decreases as the Reynolds number ofthe inert gas decreases. When the inert gas Reynolds num-ber is small enough, the surface temperature of the fiberbecomes essentially independent of the convective heattransfer between the fiber and the inert gas. Fig. 4a alsoshows that the effect of the inert gas Reynolds numberon the fiber surface temperature is restricted in the drawingregion, namely, the temperature decay in the cooling regionis hardly affected by the Reynolds number of the inert gas.

Fig. 4b shows the centerline dopant concentration forthree different Reynolds numbers of the inert gas. As seen,the inert gas Reynolds number has an interesting effect onthe centerline dopant concentration. As the Reynolds num-ber of the inert gas increases, the centerline dopant concen-tration decreases at first, but starts increasing when theinert gas Reynolds number is increased further. This isdue to the complex effect of the inert gas Reynolds numberon the temperature field. As the Reynolds number of theinert gas increases, the centerline temperature of the fiberalso decreases, but the temperature difference between thesurface and the centerline may decrease or increase depend-ing on the axial location of the fiber, which again, affectsthe dopant diffusion. Fig. 4c represents the refractive indexprofile for three different inert gas Reynolds numbers. Lit-tle difference exists for Rea = 60 and Rea = 300, while themaximum refractive index difference between the coreand the cladding is higher for Rea = 600.

3.3. Effects of cooling gas Reynolds number

The effect of cooling gas Reynolds number is investi-gated using three different values of 100, 500 and 1000.As seen in Fig. 5a, the effect of cooling gas Reynolds num-ber on the fiber surface temperature field is restricted to theforced convective cooling region, which suggests that theheat transfer in the cooling region is dominated by advec-tion due to the large speed and relatively low thermal con-ductivity of the fiber. As the cooling gas Reynolds numberincreases, the fiber surface temperature decay becomesrapid due to an increase in the local Nusselt number, butas the cooling gas Reynolds number increases further, tem-perature decay starts decreasing. This can be explained asfollows: although the local Nusselt number increases withthe cooling gas Reynolds number, the temperature of theadjacent cooling gas also rises, which results in an overalleffect of decreasing heat transfer, and therefore, a decreasein the surface temperature decay. For the cases studied, asseen in Fig. 5b and c, the effects of cooling gas Reynoldsnumber on the centerline dopant concentration and the


final refractive index profile are very small. When the cool-ing gas Reynolds number increases from 100 to 1000, thecenterline dopant concentration increases slightly, as doesthe maximum refractive index difference between the coreand cladding of the fiber. No distinguishable differenceexists in centerline dopant concentration and final refrac-tive index profile when the cooling gas Reynolds numbervaries from 100 to 500.

Fig. 5. Effects of cooling gas Reynolds number on (a) preform/fibersurface temperature, (b) preform/fiber centerline dopant concentrationdecay, and (c) fiber refractive index profile.

3.4. Effects of furnace radius

Fig. 6 shows the effect of furnace radius on the surfacetemperature (Fig. 6a), centerline dopant concentration(Fig. 6b), and refractive index profile of the fiber(Fig. 6c). Three different dimensionless furnace radii, 300,400, and 500, are examined. When the furnace radiusincreases, the view factor of the fiber with respect to the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

RF = 300

RF = 400

RF = 500

Sur

face

Tem

pera

ture

, θ


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Dop

ant C

once

ntra

tion,

C

Dimensionless Length, β(b)

1.458

1.459

1.460

1.461

0.0 0.2 0.4 0.6 0.8 1.0

Ref

ract

ive

Inde

x, n


Fig. 6. Effects of furnace radius on (a) preform/fiber surface temperature,(b) preform/fiber centerline dopant concentration decay, and (c) fiberrefractive index profile.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

θ F

= 1.1053

θ F

= 1.2105

θ F

= 1.3158

Sur

face

Tem

pera

ture

, θ


1.2


furnace is decreased slightly. This results in a smaller radi-ation transport between the furnace and the preform/fiber,which, in turn, decreases the surface temperature of thepreform/fiber. The volumetric flow rate of the inert gas alsoincreases because of the larger furnace radius, enhancingthe convection transport between the preform/fiber andthe inert gas. This again results in a decrease in the surfacetemperature of the preform/fiber. However, the magnitudeof the effect of furnace radius on the surface temperature isvery small in the cases studied, as seen in Fig. 6a. Fig. 6bshows that the centerline dopant concentration decaydecreases with the furnace radius, due to a decrease ofthe diffusion coefficient resulting from a reduced fiber tem-perature. Fig. 6c shows that the maximum refractive indexdifference between the core and cladding of the fiberincreases with the furnace radius because of the decreaseddopant diffusion.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Dop

ant C

once

ntra

tion,

C


1.465

1.470

1.475

1.480

efra

ctiv

e In

dex,

n

3.5. Effects of furnace wall temperature

In order to investigate the effect of furnace wall temper-ature, three different levels of constant furnace wall temper-ature were examined, i.e., 2100 K, 2300 K and 2500 K, or1.1053, 1.2105 and 1.3158, in dimensionless form. Asshown in Fig. 7a and b, the furnace wall temperature hasa strong effect on the temperature and dopant concentra-tion distribution of the fiber. As the furnace wall tempera-ture increases, Fig. 7a shows that the surface temperatureof the fiber increases due to the increased radiation trans-port between the preform/fiber and the furnace wall. Thisresults in an increase in the dopant diffusion, which is rep-resented by greater decay in the centerline dopant concen-tration as seen in Fig. 7b. Consequently, Fig. 7c indicatesthat the gradient of refractive index within the core ofthe fiber decreases. When the furnace wall temperature isvery low, the dopant diffusion becomes very small, andthe final refractive index of the fiber takes essentially thesame profile as that of the preform.

1.455

1.460

0.0 0.2 0.4 0.6 0.8 1.0

R


Fig. 7. Effects of furnace wall temperature on (a) preform/fiber surfacetemperature, (b) preform/fiber centerline dopant concentration decay, and(c) fiber refractive index profile.

3.6. Effects of Biot number of natural convection

The effect of the Biot number of natural convective cool-ing is shown by considering three different Biot numbers,0.0027, 0.0053 and 0.0080. As shown in Fig. 8a, the effectof Biot number on temperature distribution is felt only inthe cooling region. As expected, the temperature decaysfaster with the increasing Biot number of the natural con-vective cooling. Fig. 8b shows the effect of Biot number onthe dopant concentration; as the Biot number increases, thedopant concentration also increases due to a decrease inthe dopant diffusion due to the increasing Biot number.The maximum refractive index difference between the coreand the cladding increases with the Biot number of the nat-ural convective cooling because of the reduced dopant dif-fusion, as seen in Fig. 8c.

3.7. Effects of cooling channel radius

Fig. 9 shows the effect of cooling channel radius forthree different dimensionless cooling channel radii, i.e.,100, 200, and 400. Fig. 9a shows that as the cooling channelradius decreases, the temperature gradient at the surfaceincreases resulting in greater temperature decay near theentrance of the forced cooling. This effect is reversedtoward the end of the cooling channel due to a decreased

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

Bi = 0.0027Bi = 0.0053Bi = 0.0080

Sur

face

Tem

pera

ture

, θ


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Dop

ant C

once

ntra

tion,

C


1.458

1.459

1.460

0.0 0.2 0.4 0.6 0.8 1.0

Ref

ract

ive

Inde

x, n


Fig. 8. Effects of Biot number of natural convective cooling on (a)preform/fiber surface temperature, (b) preform/fiber centerline dopantconcentration decay, and (c) fiber refractive index profile.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

Rc = 100

Rc = 200

Rc = 400

Sur

face

Tem

pera

ture

, θ


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Dop

ant C

once

ntra

tion,

C


1.458

1.459

1.460

0.0 0.2 0.4 0.6 0.8 1.0

Ref

ract

ive

Inde

x, n


Fig. 9. Effects of cooling channel radius on (a) preform/fiber surfacetemperature, (b) preform/fiber centerline dopant concentration decay, and(c) fiber refractive index profile.


temperature gradient resulting from the temperature rise inthe cooling gas. The overall effect of cooling channel radiuson the temperature distribution is seen to be small withinthe range studied. Although cooling channel radius hassome effect on the temperature distribution in the forcedconvective cooling region, its effect on the dopant diffusionis minimal as noted in Fig. 9b. Since the dopant diffusioncoefficient is very small for low temperatures, which isthe case for the forced convective cooling region, the dop-

ant diffusion is hardly affected by the cooling channelradius. Similarly, the final refractive index profile of thefiber is also relatively unaffected by the cooling channelradius, as seen in Fig. 9c.

3.8. Effects of cooling gas thermal conductivity

The effect of cooling gas properties is investigatedby considering three different cooling gas thermal

Fig. 10. Effects of cooling channel thermal conductivity on (a) preform/fiber surface temperature, (b) preform/fiber centerline dopant concentra-tion decay, and (c) fiber refractive index profile.


conductivities, 0.012, 0.040, and 0.068, in dimensionlessform defined as Kc;f ¼ kc

kf. As the cooling gas thermal con-

ductivity increases, the surface temperature decay in thefiber increases. This is because the thermal diffusion withinthe cooling gas is much stronger for larger cooling gas ther-mal conductivity. This suggests that different cooling fluidscan be used to control the cooling rate. Although coolinggas thermal conductivity has a significant effect on the sur-

face temperature decay in the forced convective region, itseffect on the dopant diffusion is very small due to the smalldopant diffusion coefficient at low temperatures. As seen inFig. 10b, decreasing the cooling gas thermal conductivityincreases the centerline dopant concentration decayslightly; however, the centerline dopant concentrationdecay remains unchanged when the cooling gas thermalconductivity is decreased further. This analysis also appliesto the final refractive index profile of the fiber, whoseresults are shown in Fig. 10c.

3.9. Effects of dopant Peclet number

The effect of dopant properties is examined by consider-ing three different dopant Peclet numbers, 3.64 · 109,7.28 · 109, and 3.62 · 1010, and the results are shown inFig. 11. Since it has been assumed that the flow and tem-perature fields are not affected by the dopant, the surfacetemperature is independent of the dopant Peclet number,as seen in Fig. 11a. Fig. 11b shows that the centerline dop-ant concentration decay increases with the dopant Pecletnumber, as expected. The maximum refractive indexbetween the core and the cladding of the fiber increaseswith the dopant Peclet number as shown in Fig. 11c; con-sequently, the core radius of the fiber decreases with dop-ant Peclet number. This suggests that different dopantcan be used to achieve the desired refractive index profile.

The results in this section provide a comprehensive anal-ysis of the effects of various nondimensional groups on theevolution of the fiber surface temperature, dopant concen-tration, and index of refraction profile during the opticalfiber manufacturing process. The results could be used asguidelines in analyzing and designing for the final indexof refraction profile while simultaneously satisfying otherquality constraints. In a future work, optimum parametercombinations for specific design specifications will be deter-mined by integrating the present model with a numericaloptimization procedure. In the present study, the neck-down shape of the preform/fiber is approximated as a spec-ified profile in the interest of minimizing computationaleffort; a more accurate neck-down shape during a drawingprocess could be solved for by performing a force balanceat the preform/fiber surface. Further simplification of thenumerical computation resulted from a decoupling of thetemperature and velocity fields from the dopant concentra-tion in the solution procedure. It was assumed that pres-ence of dopant does not alter the thermophysicalproperties significantly, while the temperature, velocityand dopant concentration fields need to be solved togetheras a conjugate problem for an exact simulation. An effec-tive use of physics-based models for practical design callsfor incorporation of the inherent parameter uncertaintyin a manufacturing environment into the simulations. Tothis end, stochastic modeling and analysis under uncer-tainty is presently being developed and will be reportedin a forthcoming publication.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

Pe = 3.64 x 1010

Pe = 7.28 x 109

Pe = 3.64 x 109

Sur

face

Tem

pera

ture

, θ

Dimensionless Length, β (a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Dop

ant C

once

ntra

tion,

C

Dimensionless Length, β (b)

1.456

1.458

1.460

1.462

1.464

1.466

1.468

1.470

1.472

0.0 0.2 0.4 0.6 0.8 1.0

Ref

ract

ive

Inde

x, n

Dimensionless Radius, η (c)

Fig. 11. Effects of dopant Peclet number on (a) preform/fiber surfacetemperature, (b) preform/fiber centerline dopant concentration decay, and(c) fiber refractive index profile.


4. Conclusions

In this study, a detailed numerical investigation on theflow, heat and mass transfer phenomena in the drawingand cooling process of glass optical fibers has been carriedout using a finite difference approach based on primitivevariables. A prescribed neck-down profile based on exper-imental results in the literature has been used, and a coor-

dinate transformation is employed in the drawing process,with the use of covariant curvilinear velocity.

Different nondimensional parameters, namely theReynolds numbers of fiber, inert and cooling gas, radiusof furnace and cooling channel, furnace wall temperature,Biot number of natural convective cooling, cooling gas ther-mal conductivity and dopant Peclet number, were varied toinvestigate their effects on the thermal and mass transportduring the drawing and cooling process of optical fiber.Based on the results and discussion presented above, it isfound that the fiber Reynolds number, the furnace walltemperature and the dopant Peclet number have significantinfluence on the temperature field, dopant concentrationdistribution and refractive index profile, while the Reynoldsnumbers of inert and cooling gas, radius of furnace andcooling channel, Biot number of natural convective coolingand cooling gas thermal conductivity have minor effects.The results provide valuable insight for optimization ofthe process for tailored index of refraction profiles.

Acknowledgements

The work reported in this paper was funded by theNational Science Foundation Information TechnologyResearch (ITR) program through Grant No. CTS-0112822.We gratefully acknowledge the support. The authors alsoacknowledge the help of Dr. Andryas Mawardi in the prep-aration of the manuscript.

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[2] H.M. Presby, D.M. Marcuse, W.G. French, Refractive-index profil-ing of single-mode optical fibers and preforms, Appl. Phys. 18 (23)(1979) 4006–4011.

[3] J. Kirchhof, P. Kleinert, A. Funke, B. Knappe, H.R. Muller, Aboutthe smoothing of concentration fluctuations by diffusion in the glasssystems SiO2/GeO2 and SiO2/GeO2/B2O3, Cryst. Res. Technol. 22(1987) K105–K108.

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Documents

Numerical study on the dopant concentration and refractive index profile evolution in an optical fiber manufacturing process