Computational Model for Predicting the Location of Glass Solidification in Optic Fiber Drawing

Transcription

1 Proceedings of HT-FED4 4 ASME Heat Transfer/Fluids Engineering Summer Conference July -5, 4, Charlotte, North Carolina USA HT-FED4-565 Comutational Model for Predicting the Location of Glass Solidification in Otic Fiber Drawing Zhiyong Wei* and Kok-Meng Lee The George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology, Atlanta, GA gte384w@mail.gatech.edu, kokmeng.lee@me.gatech.edu *Tel: (44)894-64; Fax: (44) ABSTACT This aer resents a comutational model for redicting the location at which the glass fiber solidifies during a highseed drawing rocess. Although modeling of the otic fiber drawing rocess has been of interest for the ast two decades, traditional fiber drawing rocess uses small diameter reforms and low draw seeds, where the glass usually solidifies and turns into fiber inside the furnace. Much larger reforms drawn at higher seeds hae been used in the state-of-the-art fiber drawing systems to imroe roduction efficiency and reduce cost. Insulated ost-chambers are often added below the furnace to reduce the glass cooling rate so that the otical loss in the fiber is low. To roide a basis for design otimiation of the ost-chamber, we hae soled the conjugate roblem of the glass free surface flow and the air conection to determine the location where the glass solidifies. As radiation is the dominant mode of heat transfer in the glass, the radiatie transfer equation (TE) is soled directly by discrete ordinate method (DOM). The heat flux due to the mixed conection of the air is also numerically calculated along the glass free surface, which inoles the boundary layer flow around a continuously moing fiber and the buoyancy drien flow through the oen-ended channel. The calculated free shaes are comared against the exerimentally measured data to erify the comutational model. Keywords: otic fiber drawing, comutational, solidification. INTODUCTION To meet the growing demands of otic fibers in (the long haul and the last-mile access) networking alications, fiber manufacturers are seeking methods to draw fibers at high seeds from large reforms (glass rods) in order to imroe roduction efficiency, fiber quality, and sae manufacturing cost. Figure shows a tyical modern fiber-drawing rocess. The reform is moed slowly into a cylindrical furnace and heated aboe its melting temerature (about 58ºC for fused silica). The reform becomes soft and melts to form a neckdown rofile inside the furnace. The glass cools slowly in the insulated ost-chamber during its gradual solidification before leaing the iris (a small oening) at the bottom of the ostchamber; the slow cooling is essential to reduce otical losses in Zhi Zhou and Siu-Ping Hong OFS Norcross, GA 37 hihou@ofsotics.com, siuinghong@yahoo.com the final roduct so that the quality secifications can be met. Drien by buoyancy force, the ambient air may enter the chamber through the iris and exit the furnace at the to ring oening. The seed of the glass flow in most art of the ostchamber increases drastically (as high as 5 times) from the glass seed at the furnace exit. A boundary layer of air starts to deelo along the continuously moing glass cylinder. Our interest is to redict the location of the fiber solidification and the corresonding temerature gradient of the glass fiber for the otimal design of the drawing rocess and the ost-chamber. This requires soling the conjugate roblem of the temerature couled glass free surface flow and the mixed conection of the air inside the oen-ended chamber. Preform Fiber Air r D f f Figure Schematic of fiber drawing rocess Modeling of the high-seed fiber-drawing rocess with large-diameter erforms is challenging in that the free surface flow is strongly couled with the radiatie transfer in the glass and the air conection at the glass surface. During the late 97 s and the 98 s, arious asects of the otical fiber draw roblems were studied by a number of researchers, which L f L Furnace Postchamber Postchamber Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://

2 include Paek and unk [], Homsy and Walker [], Myers [3], and Vasilje et al [4]. These studies rimarily based on D model. Two-dimensional (D) models hae been deeloed in more recent studies. Based on the radially lumed axial elocity and the axial force balance, Choudhury et al. [5] obtained a correction scheme to generate the neck-down rofile while soling the D energy and momentum equations. Xiao and Kaminski [6] soled the D conjugate roblem using the commercial finite element code FIDAP. They studied a 5cm diameter reform and it was shown that the glass elocity distribution is close to D while the glass temerature has a D distribution. Other D studies include Lee and Jaluria [7], Choudhury and Jaluria [8], Yin and Jaluria [9] and Cheng and Jaluria []. The aboe studies only considered the furnace domain, and hae based on either one of the two assumtions; the fiber solidifies within the furnace, or an arbitrary glass diameter is secified at the furnace exit. These assumtions are only alid for drawing small diameter reforms at a low seed. The gas flow in the conjugate transort during the fiber drawing rocess has been numerically soled by a number of researchers (Choudhury et al. [], Choudhury and Jaluria [], Xiao and Kaminski [6]), where both aiding and oosing (laminar and incomressible) gas flows with a gien flow rate were considered. Howeer, the effects of the buoyancy drien oen-ended channel flow (along with temerature-deendent hysical roerties) hae not been considered in [6] [] []. In modern drawing rocesses (Figure ), the air boundary layer flow deeloed along the continuously moing fiber may turn into turbulent in a long ost-chamber, where the effects of turbulence on the fiber cooling should be inestigated. In order to redict the location of the glass solidification in modern fiber drawing rocesses, where large diameter reforms and high draw seeds are used, we consider here both the furnace and the ost-chamber in soling the conjugate roblem of the temerature couled free surface glass flow and the mixed conection of the air. Due to large temerature ariations, we account for the temerature deendency of the air roerties. Unlike reious studies [][5] where comarisons were made against exerimental data obtained for a small-diameter reform ( cm) drawn at a slow seed ( m/s), we comare our rediction against exerimentally measured neck-down rofile drawn at 8 m/s. The ost-chamber wall temerature and fiber temerature were also measured for erifying the calculation. The model resented here can be alied to the design otimiation of a high-seed draw rocess.. ANALYSIS The following assumtions are made in the formulation:. The glass flow is Newtonian and incomressible.. The surface tension at the glass free surface is neglected since it is seeral orders of magnitude smaller than the iscous force. 3. The glass is semitransarent to radiation in the range <λ<5µm and is almost oaque beyond 5µm. The scattering of radiation can be neglected (Viskanta, [3]). Based on the fact that the surface of the melting reform may undergo way hydrodynamic instabilities with the magnitude in the order of radiation waelengths, the inner and outer surfaces at the glass/air interface are treated as diffuse for the radiation reflection and transmission. 4. Preious D comutations in the furnace domain [5] [6] [] hae shown that the flow in the neck-down region is close to D for reforms u to 5cm in diameter, while the temerature distribution is D. As a first attemt to redict the fiber solidification, we use a D glass flow model with a D thermal model. The radially lumed temerature is used to obtain the glass iscosity. This semi-d model for the glass greatly reduces the comutational cost.. D Heat Transfer Model The highly iscous glass flow is strongly couled with the heat transfer through the temerature-deendent iscosity. We sole two axisymmetric thermal transort roblems for the glass temerature: the radiation transfer in the semitransarent glass and the mixed conection of the air in the oen-ended chamber. Glass domain The glass energy equation is gien by T ρc + T = ( k T + q ) + µ Φ () t where is the elocity ector; T is the temerature; ρ, C, k, µ are the density, the secific heat, the thermal conductiity and the iscosity, resectiely; q is the radiatie heat flux in the glass domain; and Φ is the iscous dissiation function. The boundary conditions for the energy equation are gien by T at r = = () T at = and L = (3) at = () k n T = q rad, oa + qcon (4) where n is the unit normal along the free surface; q rad,oa is the net radiation heat flux in the oaque band; and q con is the conectie heat flux from the air. adiation is the dominant mode of heat transfer in the neckdown region. The diergence of q can be obtained by πκ ( ) κ ( ) λ = 4 n λ I b λ T λ I λ π λ r, s dω d Ω= 4 where the sectral radiation intensity ( r,s) q (5) I λ is a function of the osition ector r, orientation ector s, and waelength λ ; I bλ T is the sectral blackbody radiation intensity gien by ( ) Planck s function; Ω is the solid angle; κ λ is the sectral absortion coefficient; and n λ is the sectral index of refraction. The radiation intensity is obtained by soling the radiatie transfer equation (TE): s Iλ ( r, s) = κλ [ nλ Ibλ ( T ) Iλ ( r, s) ] (6) In this study, the discrete ordinate method (DOM) is used to sole the TE which is transformed in a fully conseratie form in a general curilinear coordinate system. An enclosure analysis for the radiation exchange among the glass outer surface, the furnace wall and the to and the bottom disk oenings is carried out to obtain the irradiations on the free surface. These calculated irradiations are used to obtain q rad,oa and the boundary radiation intensities for the glass domain. A detailed rocedure of the method along with the boundary conditions and the fused silica radiation roerties can be found in Wei et al. Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://

3 [4] excet that the intensities from the outside of the chamber at the to and bottom of the glass domain are neglected. Air domain The glass temerature boundary condition along the free surface requires the solution of q con, which is significant in the ost-chamber due to the high surface area to olume ratio and the high moing seed of the glass. We obtain q con by numerically soling the following D continuity, momentum and energy equations for the air temerature field of the mixed conection in the furnace and ost-chamber. ρ + ( ρ) = (7) t T ρ + = + [ ( µ + µ T )( + )] + ρg (8) t T ρ C + T = [ ( k + kt ) T ] (9) t where is the ressure; g is the graity acceleration ector; and µ T and k T are the turbulent iscosity and conductiity. The algebraic mixing-length turbulence model with the emirical correlation of eddy iscosity is used to obtain µ T in Equation (8). The classical an Driest daming function (an Driest [5]) is used for the law-of-the-wall inner region, while for the wake-like log-law one, the following correlation of eddy iscosity (Luetow and Leehey, [6]) is used: µ T = cρδuτ () where c is an emirical constant and δ is the local boundary layer thickness. The friction elocity U τ is calculated using a momentum integral relation for an axisymmetric boundary layer resented by Luetow, et al. [7]. The eddy conductiity k T is then obtained using µ T and the unit turbulent Prandtl number. The boundary conditions for the air domain are gien by T at = = = =, = () at r = () u = ug, = g, T = Tg () at r = fur T fur < L f u = =, T = (3) To > L f at = L and r < oe T/ = ( > ) = =, (4) T = T ( < ) oe = ρ, = rdr f oe f at = L u = =, T = T and r > iris (5) oe where fur, oe and f are the radii of the furnace, the iris oening, and the fiber, resectiely; is the ambient ressure; T fur and T o are the furnace temerature and the ost-chamber temerature, resectiely; T amb is the ambient temerature; and T iris is the iris temerature. No-sli boundary condition has been used along the channel surfaces. The Bernoulli Equation is used to obtain the ressure at the inlet cells at the iris oening. The goerning equations for the air domain are first transformed into the curilinear coordinate system, and then amb discretied on a staggered grid. A nd order uwind scheme is used to discretie the conectie term. The Naier-Stokes equations are soled using the time marching scheme and the Pressure-Imlicit with Slitting of Oerators (PISO) algorithm (Issa, [8]).. D Glass Flow Model The D steady state glass flow model for the free surface fiber drawing rocess can be deried using the mass conseration and ertical force balance as follows: d d ( ) = π (6) d ( π ) ( σ π ) ρgπ d ρ = + (7) d d where is the axial elocity; is the glass radius which is a function of ; σ is the axial normal stress; and g is the graity acceleration. For a Newtonian fluid, the normal stress is gien by σ = + µ (8) d The ressure can be obtained from the normal force balance condition at the free interface: a = µ n ( n ) µ an ( n a ) (9) where the subscrit a denote the air side. Using Equation (6), (8) and (9) and the kinematic condition of the free surface, we hae σ = 3µ () d The detailed deriation of Equation () is gien in the aendix. The aboe result is consistent with the elongational assumtion made by Paek []. Although a similar free surface correction scheme has been suggested by Choudhury et al. [5], we reise the deriation which leads to a somewhat different solution. The boundary conditions for () is gien by at = = (a) at = L f + L = f (b) where is the reform feed rate; and f is the fiber drawing seed. Equation (b) is only alid when the glass solidifies inside the ost-chamber. If the glass temerature is aboe the melting oint and the elocity continues to increase at the exit of the ost-chamber, the ost-chamber should be lengthened so that the glass solidifies inside the chamber to minimie otical losses. From the mass conseration or Equation (6), we hae the following relationshi: = () where is the reform radius. Substituting Equation () and () into the inertia and iscous terms resectiely in Equation (7), and then integrating the resulting equation twice with resect to, we yield the following exression for : ( ) = 3µ ρg 3µ ρ d d 4 3 d d + C 3µ d + C (3) 3 Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://

4 where the integration constants are obtained by substituting the boundary conditions into the aboe equation: C = (4a) C = L ( µ ) d f ρg + 3 L µ d d (4b) 4 ρ L + d 3 µ where L = L f +L. The results in Equation (3) and (4) are different from those deried by Choudhury et al. [5] which misses an exression in the denominator for C. An additional aantage in our deriation is the simler exression for σ. Gien the axial elocity distribution (), the glass free surface rofile can be determined from Equation (): ( ) = (5) ( ) Since the calculation of () in Equation (3) requires the information of (), both and are soled iteratiely. 3. STEADY STATE SIMULATION A MATLAB rogram with C++ subroutines has been written to simulate the free surface of the draw rocess (Figure ), where the furnace temerature was measured using a M9 single color infrared thermometer (MIKON, Inc.). The reform radius is.45m and the furnace length is.45m; other alues used in the simulation are listed in Table. Table : Parameters used in the simulation f (µm) 6.5 f (m/s) 8 T fur,max (K) 4 fur /.33 L /L f 6. T fur,min (K) 7 ost /.33 The correlation obtained by Bansal [9] is used here to calculate the glass iscosity. ( T ) =.ex( /T ) µ (6) Other hysical roerties of the glass (fused silica) are taken from Fleming []. The emirical constant c in Equation () has been determined by Luetow and Leehey [6] to be.74. During the furnace temerature measuring, the thermometer measures the radiation intensities of the furnace wall at the.65µm waelength. Taking the multile reflections in the furnace chamber into account, an enclosure analysis is carried out to obtain the emissie intensities and consequently, the temerature of the furnace wall. The temerature distribution is arabolic with the maximum at the middle and minimum at both ends. Since the thermometer cannot reach measure region beyond the furnace, the ost-chamber temerature T o is calculated by balancing the heat flux at the inner wall: T T o T a amb ε [ Eb, ( T o ) H ] + k a + = (7) r= wall where ε, E b, and H are the emissiity, block-body emissie ower and irradiation at the ost-chamber inner wall; i is the ost-chamber wall inner radius; and wall is the thermal resistance between T o and T amb, which can be obtained using the i conductiity and thickness of the wall and the heat transfer coefficient at the outer wall. Non-uniform grids were used so that grids near the free interface hae a dense sacing; esecially on the air side, the dimension of the first grid adjacent to the fiber surface is made at least less than half of the fiber radius (3µm) in order to account for the shar gradients in the boundary layer. Based uon a grid sie study, the grid numbers of 5 and 6 34 (in and r directions, resectiely) are used in the glass and the air domain, resectiely. 3. Comutation Algorithm The steady-state free-surface of the glass, along with the elocity and temerature fields, has been soled as follows: Ste : Inut the assumed initial free surface rofile () and the alues of the rimitie ariables () and T(r, ). Ste : Conjugate temerature iteration (with a gien free surface and the axial elocity distribution). a) The radial elocity comonent is gien by u =.5r( / d) while satisfying the continuity equation. b) Calculate the iew factors. c) Sole the mixed conection roblem in the air domain for the heat flux along the free interface, q con = k a ( n T ) a. d) Sole the TE, then calculate q using Equation (5). e) Sole Equation () using imlicit time marching scheme. f) eeat Ste (d) until a steady state is reached. g) eeat Ste (c) until the glass temerature does not ary between two consecutie iterations. Ste 3: Free surface comutation (with the temerature field obtained in Ste ). a) Calculate the glass iscosity using Equation (6) with the radially lumed temerature gien by fur T ( ) = ( ) ( ) T r, rdr. ( ) fur b) Calculate () using Equations (3) and (4). c) Udate the free surface rofile using Equation (5). d) eeat Ste 3(b) until the free surface rofile does not change between two consecutie iterations. Ste 4: egenerate the D curilinear grid, and then reeat Ste until the relatie change between two consecutie comuted free surface rofiles at Ste 4 is less than -5. The aboe rocedures effectiely reduce the ariables to be soled during the comutation by decouling the temerature iteration and the free surface iteration. This results in a more robust and faster conergence in the comutation. 3. Model Validation In order to alidate the numerical model, the steady-state free-surface rofile of the glass rod was exerimentally measured. In the exeriment, the reform was taken out of the furnace in a short time (less than minute) in order to reent shae deformation due to the change in iew factors. After the glass cooled down, the neck-down rofile was measured by a laser scanner. Figure comares the redicted free surface rofile against the exerimentally measured data drawn at 8m/s. As shown in the figure, the redicted free surface is in excellent agreement with the exerimental measurement. 4 Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://

5 The ost-chamber inner wall temerature was measured at seeral locations using thermo-coules, and comared against the calculated alues in Figure 3, which are in good agreement. The fiber temerature at the exit of the ost-chamber, which was measured by an infrared thermo camera, was around 4K while the corresonding simulation result is 48K. This close match (with less than.5% differences) alidates the calculation of the mixed conection of the air around the fiber. glass/fiber temerature decreases more slowly in the ostchamber under a higher draw seed. This imlies that the effect of the aection increase on the glass temerature is much stronger than that of the air conection increase as the eynolds number associated with the draw seed is higher. Figure Model alidations through free surface rofiles Figure 3 Post-chamber temerature alidation 3.3 Simulation esults For a gien furnace/ost-chamber configuration, an effectie way to increase roductiity is to draw the fiber at higher seeds. Howeer, it is essential that the fiber solidifies before leaing the ost-chamber. The effects of using higher draw seeds on the fiber at steady state were studied numerically using the alues gien in Table for two draw seeds, f and f. Some obserations are briefly summaried as follows: The high draw seed has a tendency to increase the air conectie flows; it is essential to account for the effects of the mixed air conection on the location of glass solidification in the design of a ost-chamber. As shown in Figure 4, where the heat flux of the air conection and radiation along the glass free surface are comuted, the air conection is negligible in the furnace but significant in the ost-chamber as comared to the radiation. Figure 5(a) shows the redicted neck-down free surface rofiles normalied to the reform radius. When the draw seed is increased, the neck-down rofile is shifted or stretched downward to a certain extend. It is exected that the free surface region is longer for a higher draw seed. Figure 5(b) shows the axial glass temerature distributions normalied to the glass melting oint (58ºC). The Figure 4 Heat fluxes along the free surface Figure 5(c) shows the conergence of the glass diameters to that of the fiber when the glass solidifies in the ost-chamber. As shown in the figure, the fiber solidification is rather a gradual monotonically rocess than those of crystalline materials with a single hase transition temerature. Solid state glasses and fibers are actually non-crystalline suercooled liquid-like melts. We define the location where the glass reaches within a bound of.5% about the steady state diameter (5µm) as the fiber solidification location. Figure 5(d) comares two methods of locating the fiber solidification for different draw seeds; namely, constant fiber diameter, and fiber melting temerature (58ºC). The close match between the two methods shows that the glass iscosity at the melting oint is high enough that the glass behaes like a solid. Hence, the glass (melting) temerature can be reasonably used to locate the fiber solidification. Steady state solutions obtained for both seeds (as shown in Figure 5) imly that the fiber can be steadily drawn as configured. Howeer, if the draw seed increases aboe.5 f, the fiber may solidify outside the ost-chamber and thus, the trade-off for drawing at a higher seed than configured is that it requires a longer ost-chamber. 4. CONCLUSIONS The comutational model for redicting the location where the glass solidifies inside a ost-chamber during a high-seed, continuous fiber drawing has been resented. Secifically, we hae soled the conjugate roblem of the glass temerature couled with the mixed air conection around the fiber in both the furnace and ost-chamber domains for the free surface geometry of the glass flow. The solution, which has been obtained by soling the D TE directly using the DOM method and a steady-state free-surface udating scheme, has been alidated by comaring the redicted free surface to an exerimentally measured rofile drawn at 8m/s. Our results show that the location of glass solidification determined by the glass diameter is consistent with that determined by locating the glass temerature equal to its melting temerature (58ºC). The numerical model can be alied to the design otimiation 5 Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://

6 of the furnace/ost-chamber configuration of a glass fiber drawing rocess. (a) Neck-down rofiles (b) Axial temerature distributions (c) Fiber solidifications (d) Fiber solidification locations Figure 5 Effects of draw seed ACKNOWLEDGEMENT This research has been funded by OFS/Lucent Technologies. The authors would like to acknowledge Dr. Shunhe Xiong, Dr. Henry Garner and Dr. Junjun Wu in OFS for roiding the temerature measurement data and many aluable discussions EFEENCES. Paek, U. C., and.b. unk, 978, Physical Behaior of the Neck- Down egion during Furnace Drawing of Silica Fibers, Journal of Alied Physics, Vol. 49, Homsy, G. M. and K. Walker, 979, Heat Transfer in Laser Drawing of Otical Fibers, Glass Techn., Vol., No., Myers, M.., 989, A Model for Unsteady Analysis of Preform Drawing, AIChE Journal, Vol. 35, No. 4, Vasilje, V.N., G.N. Dulne, and V.D. Naumchic, 989, The Flow of a Highly Viscous Liquid with a Free Surface, Glass Technology, Vol. 3, No., Choudhury, S.., Y. Jaluria, S. H.-K. Lee, 999, A Comutational Method for Generating the Free-surface Neck-down Profile for Glass Flow in Otical Fiber Drawing, Numerical Heat Transfer, Part A, Vol. 35, Xiao, Z. and D.A. Kaminski, 997, Flow, Heat Transfer, and Free Surface Shae During the Otical Fiber Drawing Process, HTD Vol. 347, National Heat Transfer Conf., ASME, Vol. 9, Lee, S.H.-K. and Y. Jaluria, 997, Simulation of the Transort Processes in the Neck-Down egion of a Furnace Drawn Otical Fiber, Int. J. Heat Mass Transfer, Vol. 4, Choudhury, S.. and Y. Jaluria, 998, Thermal Transort Due to Material and Gas Flow in a Furnace for Drawing an Otical Fiber, J. Mater. es., Vol. 3, No., Yin, Z. and Y. Jaluria,, Neck Down and Thermally Induced Defects in High-Seed Otical Fiber Drawing, ASME Journal of Heat Transfer, Vol., Cheng, X. and Y. Jaluria,, Effect of Draw Furnace Geometry on High-Seed Otical Fiber Manufacturing, Numerical Heat Transfer, Part A, Vol. 4, Choudhury, S.., Y. Jaluria, T. Vaskoulos, and C.E. Polymerooulos, 994 Forced Conectie Cooling of Otical Fiber During Drawing Process, J. of Heat Transfer, Vol. 6, Choudhury, S.. and Y. Jaluria, 998, Thermal Transort due to Material and Gas Flow in a Furnace for Drawing an Otical Fiber, J. Mater. es., Vol. 3(), Viskanta,., and E.E., Anderson, 975, Heat transfer in Semitransarent Solids, Aances in Heat Transfer, Vol., Wei, Z. and K.M. Lee, S.W. Tchikanda, Z. Zhou, and S.P. Hong, 3, Effects of adiatie Transfer Modeling on Transient Temerature Distribution in Semitransarent Glass od, ASME Journal of Heat Transfer, Vol. 5, Van Driest, E.., 956, On Turbulent Flow Near a Wall, Journal of Aeronautical Science, Vol. 3, Luetow,.M. and P. Leehey, 986, The Eddy Viscosity in a Turbulent Boundary Layer on a Cylinder, Physics of Fluids, Vol. 9(), Luetow,.M., P. Leehey, and T. Stellinger, 985, The Thick, Turbulent Boundary Layer on a Cylinder: Mean and Fluctuation Velocities, Physics of Fluids, Vol. 8(), Issa,.I., 985, Solution of the Imlicitly Discretied Fluid Flow Equations by Oerator-Slitting, Journal of Comutational Physics, Vol. 6, Bansal, N.P. and.h. Doremus, 986, Handbook of Glass Proerties, Academic Press, Inc., New York.. Fleming, J.D., 964, Fused Silica Manual, Final eort for the U.S. Atomic Energy Commission, Oak idge, Tennessee, Project No. B Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://

7 APPENDIX We derie here the elongation model gien by Equation (), which eliminates the ressure in Equation (8) using the free surface boundary condition gien by Equation (9). We began with the st term on the right-hand-side of Equation (9). The unit normal along the free surface ointing outward is gien by n iˆ n ˆ r j iˆ n = + = ˆj + + (A) where î and ĵ are the unit ectors along the radial and axial directions resectiely; and the rime denotes the deriatie with resect to. Using Equation (A) and = uiˆ + ˆj, it yields dnr dn n ( n ) = nr + nrn + n + n u + n (A) d d d where u and its artial deriaties are obtained as follows. Along the free surface, we hae the steady state kinematics: u = (A3) The deriatie of Equation (A.3) with resect to is gien by = + (A4) d Using the D continuity equation / + u / r + / = at the free surface and Equation (A3), we obtain = (A5) d Substituting Equations (A) and (A3)-(A5) into Equation (A) and assuming is small (the maximum alue of in our simulation is around.3, the square of which is.53<<), we obtain the following equation uon simlification: n ( n ) = (A6) + d d Exanding Equation (6) we obtain = (A7) d Thus, the st term in the right-hand-side of Equation (9) becomes n ( n ) (A8) d The ressure term in the D momentum equation for the solution of the elocity can be eliminated on the basis of the following arguments. Since the ressure ariation on the air side can be neglected as comared to that on the glass side, the air ressure is taken as a reference and set to. Furthermore, the normal stress of the air is also negligible as comared to that of the glass. Hence, Equation (9) can be simlified to = µ n ( n ) (A9) The ressure can be eliminated from Equation (8) by using Equation (A9) along with the substitution of Equation (A8). NOMENCLATUE C Secific heat E Sectral blackbody emissie ower bλ E, Blackbody emissie ower on the ost-chamber wall b H Irradiation on ost-chamber inner wall I λ Sectral radiatie intensity I bλ Sectral blackbody intensity L f Furnace length L Post-chamber length L s Solidification location Glass radius f Fiber radius Preform radius i Post-chamber wall inner radius oe Iris oening radius wall Thermal resistance of the ost-chamber wall T Temerature T adially lumed temerature T fur Furnace temerature T o Post-chamber temerature T amb Ambient temerature U Friction elocity Velocity ector g Graity acceleration ector k Thermal conductiity n λ Index of refraction n Unit normal ector Pressure q adiation heat flux q rad,oa Net radiation heat flux in the oaque band q con Conectie heat flux from the air r, adial and axial coordinates r Position ector s Orientation ector u, adial and axial comonents of elocity d, f Draw seed, f = 8m/s Preform feed rate κ λ Sectral absortion coefficient ε Emissiity of the ost-chamber wall ρ Density µ Dynamic iscosity µ T Eddy iscosity Φ Dissiation function σ Axial normal stress Ω Solid angle δ Boundary layer thickness λ Waelength 7 Coyright 4 ASME Downloaded 8 Jun to edistribution subject to ASME license or coyright; see htt://