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Numerical simulation of the focused powder streams in coaxial laser cladding
Journal of Materials Processing Technology 105 (2000) 17±23
Numerical simulation of the focused powder streams in coaxial laser cladding Jehnming Lin* Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC Received 7 December 1998
Abstract The powder ¯ow structures of a coaxial nozzle for laser cladding was simulated by the numerical programme FLUENT with various arrangements of the nozzle exit. Both focused and columnar powder streams could be generated in a coaxial nozzle. It can be found that the concentration mode is in¯uenced signi®cantly by the nozzle arrangement and gas ¯ow settings. According to the numerical results, more than 50% powder concentration can be increased in a gas stream through a speci®c nozzle arrangement for coaxial laser cladding. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Numerical simulation; Coaxial laser cladding; Columnar powder streams
1. Introduction
2. Formulation of single-phase turbulent ¯ow
Coaxial laser cladding is one of the new processes applied to the rapid prototyping industry for its excellent ¯exibility in generating the metallic components directly from CAD drawings [1]. However, due to the inef®cient usage of the powder in the process, many attempts have been made to improve the nozzle design [2,3]. There are many industrial processes similar to coaxial laser cladding, which involve jets impinging on a solid surface. These vary from the impact of rocket exhausts to mixing systems, water jet cutting and shrouding systems. Many of these are discussed extensively in books on the subject [4±6]. For the speci®c problem of turbulent free jets and compound jets of more than one ¯ow stream, the ¯ow structure has been explored with agreement between theory and practice [7±9]. However, the reported literature does not include much work on compound jets containing ¯owing powder for laser cladding. This paper describes a numerical method leading to solutions of the turbulent gas±powder ¯ow problem based on a coaxial jet with an average Reynolds number (Re) of 2000 at the jet entrance [10]. A detailed numerical analysis of the powder stream without laser radiation is made by FLUENT software, which is a ®nite difference numerical algorithm based on a speci®ed control-volume approach.
A signi®cant property of turbulent jet ¯ow is that momentum, heat and mass are transferred across the ¯ow at rates much greater than those of laminar ¯ow with molecular transport processes by viscosity and diffusion. The conservation equations used for turbulent ¯ows are obtained from those of laminar ¯ows using a time averaging procedure commonly known as Reynolds averaging [7]. When the ¯ow is turbulent, the characteristic variables of pressure, velocity, and temperature may vary with both space and time.
* Tel.: 886-6-2757575; fax: 886-6-2352973. E-mail address: [email protected] (J. Lin).
3. Governing equations The mathematical model used in this work is based on the Navier±Stokes system of differential equations with the Reynolds method of averaging the time-dependent equations, together with the standard k±e turbulence model, discussed more fully below. The time-averaged, governing equations for turbulent ¯ow are expressed as follows [7,8]. Conservation of mass: @ ruj 0 @xj
(1)
where r is the density of the gas, and uj the velocity vector in the jth direction.
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 8 4 - 7
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J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
Nomenclature a CD C(l) d D Dp F FD g h k p Re t u up U0 U1 Uc x
constant drag coefficient powder concentration function by weight (kg/m3) beam diameter (m) diameter (m) particle diameter (m) force (N) drag force (N) gravitational acceleration (m/s2) convective heat transfer coefficient (W/m2 K) kinetic energy of turbulence (m2/s2) pressure (N/m2) Reynolds number time (s) flow velocity (m/s) particle velocity (m/s) inner shield gas velocity (m/s) outer shield gas velocity (m/s) carrier shield gas velocity (m/s) Cartesian coordinate
Greek a y k m r sn t tR o
letters diffusivity (m2/s) angle; stream angle thermal conductivity (W/mK) viscosity (kg/s m) density (kg/m3) Prandtl number shear stress (N/m2) Reynolds shear stress (N/m2) solid angle
(5)
mt @r rsh @xi
(6) (7) (8)
where C1e1.44, C2e1.92, sk1.0 and se1.3 are empirical constants, sh is the turbulent Prandtl number, mtCp/kt, Gk is the rate of production of kinetic energy of turbulence, and Gb is the generation of turbulence due to buoyancy [11]. (2)
(3)
where m is the molecular viscosity. dij1 for ij, otherwise dij0, and mt is the turbulent viscosity given by k2 e
Conservation of kinetic energy of turbulence: @ @ mt @k rui k Gk Gb ÿ re @xi @xi sk @xi
Gb ÿgi
where tij is the stress and gi the gravitational acceleration.
mt rCm
The modelling of turbulent ¯ows requires appropriate modelling procedures to describe the effects of turbulent ¯uctuation of velocity and the scalar quantities on the basic conservation equations. Since numerical simulations of the turbulence have been studied at a research level [6], the scope of this study is not concerned with model development, but focuses only on adopting a reliable turbulence model for a jet ¯ow problem. The standard k±e model of turbulence is used in this study to close the system of conservation equations from Eqs. (1)± (4). Two additional conditions are the equations for conservation of the kinetic energy of turbulence k and its dissipation e.
@uj @ui @ui Gk mt @xi @xj @xj
Conservation of momentum:
The stress is given by @ui @uj 2 @ui dij ÿ mt tij m mt @xj @xi 3 @xi
4. k±e model for turbulent ¯ows
Conservation of dissipation of kinetic energy of turbulence: @ @ mt @e e e2 rui e C1e Gk Gb ÿ C2e r k @xi @xi se @xi k
Subscripts g gas i, j directions in Cartesian coordinates p particle t turbulent flow 1 infinity
@ @p @tij rui uj ÿ rgi @xj @xi @xj
turbulence, and e the dissipation of kinetic energy of turbulence, which is de®ned in the k±e turbulence model as follows.
(4)
where Cm0.09 is a constant, k the kinetic energy of
5. Characteristics of particle±gas laden ¯ow Gas±particle ¯ows are characterised by coupling between phases. The coupling through heat transfer from the gas phase to the particle phase and the momentum change responsible for particle motion to the aerodynamic drag need to be incorporated in the numerical ¯ow model. The behaviour of particles suspended in a turbulent ¯ow depends on the properties of both the particles and the ¯ow. Turbulent dispersion of both the particles and the carrier ¯uid can be handled by the concept of an eddy diffusion energy in some range of the particle size distribution. The momentum transfer between the interaction of the two
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
phases has been investigated and many criteria have been proposed. Because of the complexity of the computation for particle and gas ¯ow, many techniques have been proposed, but the uncertainty of the prediction model is still large [8,10]. In order to de®ne the properties of a gas±particle mixture, the volume must be large enough to contain suf®cient particles for a stationary average. Thus, the particles could not be treated as a continuum in a ¯ow system of comparable dimensions. The use of differential equations to relate property changes of a particle cloud (and mixture) through the application of the conservation equations would be cautioned in this situation [12]. With reasonable assumptions, the gas is responsible for the particle trajectories and property changes along the trajectories. This condition is identi®ed as a one-way coupling [12,13]. The assumption of one-way coupling was used frequently in the early numerical and analytic models for gas±particle ¯ows. In this study, information is con®ned to travel in one direction only, and solutions can be obtained by integrating the governing equations in this direction [12]. 6. Particle dynamics in the gas ¯ow A two-phase ¯ow problem involving a dispersed second phase is solved with the additional transport equations of the second phase. The trajectory of a dispersed phase particle is solved by integrating the force balance on the particle in a Lagrangian reference frame (following the particle coordinate). This force balance equates the particle inertia with the force acting on the particle, and can be written as gi rp ÿ r1 dup;i FD u1;i ÿ up;i Fi dt rp
19
Equations similar to (9) and (13) are solved in each coordinate direction to predict the trajectories of the dispersed phase. Eq. (9) incorporates additional forces Fi in the particle force balance which can be important under special circumstances [11]. 7. Numerical analysis The solution technique used in this study is based on the FLUENT ®nite difference models, which solve the conservation equations for mass, momentum, energy, and chemical species by a speci®ed control-volume method [11]. The governing equations are discretised on a curvilinear grid to enable computations in complex/irregular geometry. The combination of gas streams from the central, middle, and outer streams can generate various shrouding gas structures in a coaxial nozzle for laser cladding, as shown in Fig. 1 [2]. Two nozzle arrangements were used to control and modify the shroud gas stream. As illustrated in Fig. 1, these are: (a) with the central nozzle at an inward position; (b) with the central nozzle at an outward position. The geometric domains and boundary conditions for simulating coaxial jet by the FLUENT programme are illustrated in Fig. 2, where metal powder is delivered by the mild stream with a carrier gas velocity Uc. A cylindrical coordinate system was applied to the coaxial jet problems. Only a half-plane domain is needed in the computation due to the symmetric nature of the problem.
(9)
where FD(u1,iÿup,i) is the drag force per unit particle mass and FD
18m CD Re rp Dp 2 24
(10)
where Re stands for the relative Reynolds number, which is de®ned as Re
r1 Dp jup;i ÿ u1;i j m
(11)
The drag coef®cient, CD, is a function of the relative Reynolds number of the following general form: a2 a3 (12) C D a1 Re Re2 where a's are constants which apply over several ranges of Re [11]. Integration with time of Eq. (9) yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by dxi up;i dt
(13)
Fig. 1. The con®gurations of the coaxial nozzle at position of the central nozzle at the exit: (a) inward position; (b) outward position.
20
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
8. Grid selections
Fig. 2. Geometry and boundary condition for ¯ow analysis (Uc: carrier gas±powder velocity, U0: inner gas velocity, U1: outer gas velocity).
The grid selection is an important technique to improve the accuracy of the solutions in most of the numerical simulation methods. There are two methods of grid generation that could be adopted in FLUENT. For a simple geometry without an edge or curve, the Cartesian coordinate grid system might be used. Otherwise, the body ®tted coordinate (BFC) grid system is applied. In order to accurately predict the powder concentration distribution of the coaxial jet stream, the BFC grid system was used in this study. This allowed the non-standard geometry of the nozzle to be mapped into Cartesian or cylindrical geometry [11]. The acceptability of the grid generation would be checked automatically by the FLUENT programme and from the
Fig. 3. Showing the grid selection of the FLUENT computation for the central nozzle at: (a) inward position (2919); (b) outward position (2919).
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
21
residuals list of the convergence in the computation. Fig. 3 shows the BFC grid selection for different geometric domains in the ¯ow analysis of the present study. 9. Assumptions in the computation The following are the assumptions of the jet ¯ow computed by FLUENT software: 1. The jet ¯ow problem is treated as a steady-state turbulent ¯ow with constant velocity distribution in the inlet boundary. 2. Only the forces of drag, inertia and gravity are considered in this study, other forces such as pressure and the surrounding ¯ow acceleration being neglected. 3. The effect of velocity ¯uctuation on the particle in the turbulence ¯ow is considered in FLUENT for the particle trajectory calculation. 4. A dilute gas particle stream is assumed in this simulation work. The particle collision is not considered in the FLUENT programme for solving the second phase problem. 5. Heat transfer by laser radiation is not included. 6. The particle size is assumed to follow the general Rossin±Rammler distribution expression [9].
10. Results and discussion The physical model of the compound jet with dispersed powder was solved. Results are based on different con®gurations of the nozzle with various gas settings. Normally more than 400 numerical iterations are required for each simulation case to solve the coupling problems of the gas and the powder phases. By choosing a proper boundary condition of the gas velocities (Uc for carrier gas velocity, U0 for centre gas velocity and U1 for outer gas velocity), typical numerical solutions of the contour of the particle concentration are shown in Figs. 4 and 5, where the ¯ow boundary conditions are selected as for argon gas at constant velocities (U04 m/ s and U18 m/s) on the inlet boundary and constant pressure on the outlet boundary, as illustrated in Fig. 2. A trapped wall boundary is selected for the particle phase. The total mass ¯ow rate of stainless steel powder is 0.04 g/s for a size range from 45 to 105 mm in 30 groups injected into the stream with the same spraying velocity and angle of the carrier ¯ow. It is seen that the focusability of the powder stream is signi®cantly affected by the arrangement of the nozzle exit. From the concentration results as shown in Figs. 4 and 5 with a nozzle at inward and outward positions, the mode of powder stream can be categorised into two types: columnar and focused streams. With a nozzle at the outward position as shown in Fig. 5, the powder stream will disperse widely at
Fig. 4. Contours of stainless steel powder concentration (kg/m3) for a nozzle at the inward position, for a mass ¯ow rate of 0.04 g/s, U04 m/s and U18 m/s.
the exit with a non-uniform concentration pro®le along the jet stream and its peak concentration is 0.315 kg/m3 at about 5 mm to the nozzle exit. As the nozzle moved inwards, the stream structure changes dramatically without clear focus. The peak powder concentration is 51% of the focused stream as shown in Fig. 4. It should be pointed out that a zero gradient of any scalar quantity at an axisymmetric boundary is enforced by the numerical programme [9]. Therefore, the second phase properties such as particle concentration at the stream centre should be the same as that in the ¯uid grid cell near the axis. The effect of the gas velocity settings was examined separately. The concentration pro®les in the longitudinal direction of the powder stream are plotted in Fig. 6 for various jet velocities with the nozzle at the inward position. It is seen that the stream velocity affects the powder concentration signi®cantly. The powder concentration will be
22
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
Fig. 5. Contours of stainless steel powder concentration (kg/m3) for a nozzle at the outward position, for a mass ¯ow rate of 0.04 g/s, U04 m/s and U18 m/s.
increased by decreasing the ¯ow velocity. However, the peak powder concentration is always near the nozzle exit at a distance of 5 mm for a nozzle at the inward position. In solving the focus position of the powder streams with various gas settings at the outward position, Fig. 7 shows that the peak concentration of the stream is located at a distance ranging from 8 to 14 mm from the nozzle exit. Unlike the nozzle at the inward position, a clear stream focus occurs in the streams. This is similar to the columnar streams, where a lower powder concentration occurs at the focused streams with a higher outer and inner shield gas velocities. As shown in Fig. 7, the stream focus shifts to the nozzle exit with increase in the inner and outer gas velocities. Since the compound jets mix together due to a propagation of the shearing ¯ow caused by the different velocities between the jets, in the case of a compound jet with a highvelocity centre stream, there are two shear layers which propagate towards the centre and the outer region. The increase of the inner jet momentum further diffuses the
Fig. 6. The in¯uence of the compound jet velocity on the distribution of stainless steel powder concentration along the beam axial direction at the centre of powder stream for an inward nozzle at a powder mass ¯ow rate of 0.05 g/s: (a) U02 m/s; (b) U04 m/s.
turbulent ¯ow, and the particle dispersion rate is signi®cantly enhanced by increasing the mean ¯ow velocity. Typical results are the particle concentration drop and the ¯ow ®eld expansion [9].
Fig. 7. The in¯uence of the compound jets velocity on the stream focus of nozzle at outward position at a powder ¯ow rate of 0.05 g/s.
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
23
11. Conclusions
References
With a coaxial nozzle at the inward exit arrangement, the powder stream forms a columnar structure. The powder concentration tends to reduce with increasing velocities of the gas ¯ows. A signi®cant ¯ow-mixing occurs at the exit of the nozzle at the inward position and the peak powder concentration is always located near the nozzle exit at a distance of 5 mm for the proposed nozzle design. Using a nozzle exit at the outward position, the powder stream can form a clear focused stream. The powder concentration tends to reduce with increasing values of the gas ¯ow velocities. The peak concentration of the stream focus is located at a distance ranging from 8 to 14 mm from the nozzle exit, but shifts towards the nozzle exit with increasing gas ¯ow velocities. The powder concentration in a nozzle at the inward position tends to reduce the peak powder concentration by up to 50% of that at the outward position.
[1] J.L. Koch, J. Mazumder, Rapid prototyping by laser cladding, in: ICALEO, 1993, pp. 556±559. [2] J. Lin, W.M. Steen, Design characteristics and development of a nozzle for coaxial laser cladding, J. Laser Appl. 10 (1998) 55±63. [3] P.-A. Vetter, Th. Engel, J. Fontaine, Laser cladding: the relative parameters for process control, SPIE Proc. 2207 (1994) 452±462. [4] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1979. [5] G.N. Abramovich, The Theory of Turbulent Jets, MIT Press, Cambridge, MA, 1963. [6] J.O. Hinze, Turbulence, McGraw-Hill, New York, 1975. [7] N. Rajaratnam, Turbulent Jets, Elsevier, Amsterdam, 1976. [8] R.A. Antonia, R.W. Bilger, An experimental investigation of an axisymmetric jet in a co-¯owing air stream, J. Fluid Mech. 61 (1973) 805±822. [9] J. Fan, H. Zhao, K. Chen, An experimental study of two-phase turbulent coaxial jets, Experiments in Fluids 13 (1992) 279±287. [10] V.L. Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1961. [11] FLUENT 4.4.4 User Guide, Fluent Inc., 1996. [12] C.T. Crowe, Review Ð numerical models for dilute gas±particle ¯ows, ASME Trans. J. Fluids Eng. 104 (1982) 297±303. [13] M.P. Sharma, C.T. Crowe, A novel physico-computational model for quasi one-dimensional gas±particle ¯ows, ASME Trans. J. Fluid Eng. 100 (1978) 343±349.
Acknowledgements Many thanks are due to Professor W.M. Steen of the University of Liverpool for his invaluable suggestions in this work, which is sponsored by NSC, Taiwan.
Numerical simulation of the focused powder streams in coaxial laser cladding Jehnming Lin* Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC Received 7 December 1998
Abstract The powder ¯ow structures of a coaxial nozzle for laser cladding was simulated by the numerical programme FLUENT with various arrangements of the nozzle exit. Both focused and columnar powder streams could be generated in a coaxial nozzle. It can be found that the concentration mode is in¯uenced signi®cantly by the nozzle arrangement and gas ¯ow settings. According to the numerical results, more than 50% powder concentration can be increased in a gas stream through a speci®c nozzle arrangement for coaxial laser cladding. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Numerical simulation; Coaxial laser cladding; Columnar powder streams
1. Introduction
2. Formulation of single-phase turbulent ¯ow
Coaxial laser cladding is one of the new processes applied to the rapid prototyping industry for its excellent ¯exibility in generating the metallic components directly from CAD drawings [1]. However, due to the inef®cient usage of the powder in the process, many attempts have been made to improve the nozzle design [2,3]. There are many industrial processes similar to coaxial laser cladding, which involve jets impinging on a solid surface. These vary from the impact of rocket exhausts to mixing systems, water jet cutting and shrouding systems. Many of these are discussed extensively in books on the subject [4±6]. For the speci®c problem of turbulent free jets and compound jets of more than one ¯ow stream, the ¯ow structure has been explored with agreement between theory and practice [7±9]. However, the reported literature does not include much work on compound jets containing ¯owing powder for laser cladding. This paper describes a numerical method leading to solutions of the turbulent gas±powder ¯ow problem based on a coaxial jet with an average Reynolds number (Re) of 2000 at the jet entrance [10]. A detailed numerical analysis of the powder stream without laser radiation is made by FLUENT software, which is a ®nite difference numerical algorithm based on a speci®ed control-volume approach.
A signi®cant property of turbulent jet ¯ow is that momentum, heat and mass are transferred across the ¯ow at rates much greater than those of laminar ¯ow with molecular transport processes by viscosity and diffusion. The conservation equations used for turbulent ¯ows are obtained from those of laminar ¯ows using a time averaging procedure commonly known as Reynolds averaging [7]. When the ¯ow is turbulent, the characteristic variables of pressure, velocity, and temperature may vary with both space and time.
* Tel.: 886-6-2757575; fax: 886-6-2352973. E-mail address: [email protected] (J. Lin).
3. Governing equations The mathematical model used in this work is based on the Navier±Stokes system of differential equations with the Reynolds method of averaging the time-dependent equations, together with the standard k±e turbulence model, discussed more fully below. The time-averaged, governing equations for turbulent ¯ow are expressed as follows [7,8]. Conservation of mass: @ ruj 0 @xj
(1)
where r is the density of the gas, and uj the velocity vector in the jth direction.
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 8 4 - 7
18
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
Nomenclature a CD C(l) d D Dp F FD g h k p Re t u up U0 U1 Uc x
constant drag coefficient powder concentration function by weight (kg/m3) beam diameter (m) diameter (m) particle diameter (m) force (N) drag force (N) gravitational acceleration (m/s2) convective heat transfer coefficient (W/m2 K) kinetic energy of turbulence (m2/s2) pressure (N/m2) Reynolds number time (s) flow velocity (m/s) particle velocity (m/s) inner shield gas velocity (m/s) outer shield gas velocity (m/s) carrier shield gas velocity (m/s) Cartesian coordinate
Greek a y k m r sn t tR o
letters diffusivity (m2/s) angle; stream angle thermal conductivity (W/mK) viscosity (kg/s m) density (kg/m3) Prandtl number shear stress (N/m2) Reynolds shear stress (N/m2) solid angle
(5)
mt @r rsh @xi
(6) (7) (8)
where C1e1.44, C2e1.92, sk1.0 and se1.3 are empirical constants, sh is the turbulent Prandtl number, mtCp/kt, Gk is the rate of production of kinetic energy of turbulence, and Gb is the generation of turbulence due to buoyancy [11]. (2)
(3)
where m is the molecular viscosity. dij1 for ij, otherwise dij0, and mt is the turbulent viscosity given by k2 e
Conservation of kinetic energy of turbulence: @ @ mt @k rui k Gk Gb ÿ re @xi @xi sk @xi
Gb ÿgi
where tij is the stress and gi the gravitational acceleration.
mt rCm
The modelling of turbulent ¯ows requires appropriate modelling procedures to describe the effects of turbulent ¯uctuation of velocity and the scalar quantities on the basic conservation equations. Since numerical simulations of the turbulence have been studied at a research level [6], the scope of this study is not concerned with model development, but focuses only on adopting a reliable turbulence model for a jet ¯ow problem. The standard k±e model of turbulence is used in this study to close the system of conservation equations from Eqs. (1)± (4). Two additional conditions are the equations for conservation of the kinetic energy of turbulence k and its dissipation e.
@uj @ui @ui Gk mt @xi @xj @xj
Conservation of momentum:
The stress is given by @ui @uj 2 @ui dij ÿ mt tij m mt @xj @xi 3 @xi
4. k±e model for turbulent ¯ows
Conservation of dissipation of kinetic energy of turbulence: @ @ mt @e e e2 rui e C1e Gk Gb ÿ C2e r k @xi @xi se @xi k
Subscripts g gas i, j directions in Cartesian coordinates p particle t turbulent flow 1 infinity
@ @p @tij rui uj ÿ rgi @xj @xi @xj
turbulence, and e the dissipation of kinetic energy of turbulence, which is de®ned in the k±e turbulence model as follows.
(4)
where Cm0.09 is a constant, k the kinetic energy of
5. Characteristics of particle±gas laden ¯ow Gas±particle ¯ows are characterised by coupling between phases. The coupling through heat transfer from the gas phase to the particle phase and the momentum change responsible for particle motion to the aerodynamic drag need to be incorporated in the numerical ¯ow model. The behaviour of particles suspended in a turbulent ¯ow depends on the properties of both the particles and the ¯ow. Turbulent dispersion of both the particles and the carrier ¯uid can be handled by the concept of an eddy diffusion energy in some range of the particle size distribution. The momentum transfer between the interaction of the two
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
phases has been investigated and many criteria have been proposed. Because of the complexity of the computation for particle and gas ¯ow, many techniques have been proposed, but the uncertainty of the prediction model is still large [8,10]. In order to de®ne the properties of a gas±particle mixture, the volume must be large enough to contain suf®cient particles for a stationary average. Thus, the particles could not be treated as a continuum in a ¯ow system of comparable dimensions. The use of differential equations to relate property changes of a particle cloud (and mixture) through the application of the conservation equations would be cautioned in this situation [12]. With reasonable assumptions, the gas is responsible for the particle trajectories and property changes along the trajectories. This condition is identi®ed as a one-way coupling [12,13]. The assumption of one-way coupling was used frequently in the early numerical and analytic models for gas±particle ¯ows. In this study, information is con®ned to travel in one direction only, and solutions can be obtained by integrating the governing equations in this direction [12]. 6. Particle dynamics in the gas ¯ow A two-phase ¯ow problem involving a dispersed second phase is solved with the additional transport equations of the second phase. The trajectory of a dispersed phase particle is solved by integrating the force balance on the particle in a Lagrangian reference frame (following the particle coordinate). This force balance equates the particle inertia with the force acting on the particle, and can be written as gi rp ÿ r1 dup;i FD u1;i ÿ up;i Fi dt rp
19
Equations similar to (9) and (13) are solved in each coordinate direction to predict the trajectories of the dispersed phase. Eq. (9) incorporates additional forces Fi in the particle force balance which can be important under special circumstances [11]. 7. Numerical analysis The solution technique used in this study is based on the FLUENT ®nite difference models, which solve the conservation equations for mass, momentum, energy, and chemical species by a speci®ed control-volume method [11]. The governing equations are discretised on a curvilinear grid to enable computations in complex/irregular geometry. The combination of gas streams from the central, middle, and outer streams can generate various shrouding gas structures in a coaxial nozzle for laser cladding, as shown in Fig. 1 [2]. Two nozzle arrangements were used to control and modify the shroud gas stream. As illustrated in Fig. 1, these are: (a) with the central nozzle at an inward position; (b) with the central nozzle at an outward position. The geometric domains and boundary conditions for simulating coaxial jet by the FLUENT programme are illustrated in Fig. 2, where metal powder is delivered by the mild stream with a carrier gas velocity Uc. A cylindrical coordinate system was applied to the coaxial jet problems. Only a half-plane domain is needed in the computation due to the symmetric nature of the problem.
(9)
where FD(u1,iÿup,i) is the drag force per unit particle mass and FD
18m CD Re rp Dp 2 24
(10)
where Re stands for the relative Reynolds number, which is de®ned as Re
r1 Dp jup;i ÿ u1;i j m
(11)
The drag coef®cient, CD, is a function of the relative Reynolds number of the following general form: a2 a3 (12) C D a1 Re Re2 where a's are constants which apply over several ranges of Re [11]. Integration with time of Eq. (9) yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by dxi up;i dt
(13)
Fig. 1. The con®gurations of the coaxial nozzle at position of the central nozzle at the exit: (a) inward position; (b) outward position.
20
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
8. Grid selections
Fig. 2. Geometry and boundary condition for ¯ow analysis (Uc: carrier gas±powder velocity, U0: inner gas velocity, U1: outer gas velocity).
The grid selection is an important technique to improve the accuracy of the solutions in most of the numerical simulation methods. There are two methods of grid generation that could be adopted in FLUENT. For a simple geometry without an edge or curve, the Cartesian coordinate grid system might be used. Otherwise, the body ®tted coordinate (BFC) grid system is applied. In order to accurately predict the powder concentration distribution of the coaxial jet stream, the BFC grid system was used in this study. This allowed the non-standard geometry of the nozzle to be mapped into Cartesian or cylindrical geometry [11]. The acceptability of the grid generation would be checked automatically by the FLUENT programme and from the
Fig. 3. Showing the grid selection of the FLUENT computation for the central nozzle at: (a) inward position (2919); (b) outward position (2919).
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
21
residuals list of the convergence in the computation. Fig. 3 shows the BFC grid selection for different geometric domains in the ¯ow analysis of the present study. 9. Assumptions in the computation The following are the assumptions of the jet ¯ow computed by FLUENT software: 1. The jet ¯ow problem is treated as a steady-state turbulent ¯ow with constant velocity distribution in the inlet boundary. 2. Only the forces of drag, inertia and gravity are considered in this study, other forces such as pressure and the surrounding ¯ow acceleration being neglected. 3. The effect of velocity ¯uctuation on the particle in the turbulence ¯ow is considered in FLUENT for the particle trajectory calculation. 4. A dilute gas particle stream is assumed in this simulation work. The particle collision is not considered in the FLUENT programme for solving the second phase problem. 5. Heat transfer by laser radiation is not included. 6. The particle size is assumed to follow the general Rossin±Rammler distribution expression [9].
10. Results and discussion The physical model of the compound jet with dispersed powder was solved. Results are based on different con®gurations of the nozzle with various gas settings. Normally more than 400 numerical iterations are required for each simulation case to solve the coupling problems of the gas and the powder phases. By choosing a proper boundary condition of the gas velocities (Uc for carrier gas velocity, U0 for centre gas velocity and U1 for outer gas velocity), typical numerical solutions of the contour of the particle concentration are shown in Figs. 4 and 5, where the ¯ow boundary conditions are selected as for argon gas at constant velocities (U04 m/ s and U18 m/s) on the inlet boundary and constant pressure on the outlet boundary, as illustrated in Fig. 2. A trapped wall boundary is selected for the particle phase. The total mass ¯ow rate of stainless steel powder is 0.04 g/s for a size range from 45 to 105 mm in 30 groups injected into the stream with the same spraying velocity and angle of the carrier ¯ow. It is seen that the focusability of the powder stream is signi®cantly affected by the arrangement of the nozzle exit. From the concentration results as shown in Figs. 4 and 5 with a nozzle at inward and outward positions, the mode of powder stream can be categorised into two types: columnar and focused streams. With a nozzle at the outward position as shown in Fig. 5, the powder stream will disperse widely at
Fig. 4. Contours of stainless steel powder concentration (kg/m3) for a nozzle at the inward position, for a mass ¯ow rate of 0.04 g/s, U04 m/s and U18 m/s.
the exit with a non-uniform concentration pro®le along the jet stream and its peak concentration is 0.315 kg/m3 at about 5 mm to the nozzle exit. As the nozzle moved inwards, the stream structure changes dramatically without clear focus. The peak powder concentration is 51% of the focused stream as shown in Fig. 4. It should be pointed out that a zero gradient of any scalar quantity at an axisymmetric boundary is enforced by the numerical programme [9]. Therefore, the second phase properties such as particle concentration at the stream centre should be the same as that in the ¯uid grid cell near the axis. The effect of the gas velocity settings was examined separately. The concentration pro®les in the longitudinal direction of the powder stream are plotted in Fig. 6 for various jet velocities with the nozzle at the inward position. It is seen that the stream velocity affects the powder concentration signi®cantly. The powder concentration will be
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J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
Fig. 5. Contours of stainless steel powder concentration (kg/m3) for a nozzle at the outward position, for a mass ¯ow rate of 0.04 g/s, U04 m/s and U18 m/s.
increased by decreasing the ¯ow velocity. However, the peak powder concentration is always near the nozzle exit at a distance of 5 mm for a nozzle at the inward position. In solving the focus position of the powder streams with various gas settings at the outward position, Fig. 7 shows that the peak concentration of the stream is located at a distance ranging from 8 to 14 mm from the nozzle exit. Unlike the nozzle at the inward position, a clear stream focus occurs in the streams. This is similar to the columnar streams, where a lower powder concentration occurs at the focused streams with a higher outer and inner shield gas velocities. As shown in Fig. 7, the stream focus shifts to the nozzle exit with increase in the inner and outer gas velocities. Since the compound jets mix together due to a propagation of the shearing ¯ow caused by the different velocities between the jets, in the case of a compound jet with a highvelocity centre stream, there are two shear layers which propagate towards the centre and the outer region. The increase of the inner jet momentum further diffuses the
Fig. 6. The in¯uence of the compound jet velocity on the distribution of stainless steel powder concentration along the beam axial direction at the centre of powder stream for an inward nozzle at a powder mass ¯ow rate of 0.05 g/s: (a) U02 m/s; (b) U04 m/s.
turbulent ¯ow, and the particle dispersion rate is signi®cantly enhanced by increasing the mean ¯ow velocity. Typical results are the particle concentration drop and the ¯ow ®eld expansion [9].
Fig. 7. The in¯uence of the compound jets velocity on the stream focus of nozzle at outward position at a powder ¯ow rate of 0.05 g/s.
J. Lin / Journal of Materials Processing Technology 105 (2000) 17±23
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11. Conclusions
References
With a coaxial nozzle at the inward exit arrangement, the powder stream forms a columnar structure. The powder concentration tends to reduce with increasing velocities of the gas ¯ows. A signi®cant ¯ow-mixing occurs at the exit of the nozzle at the inward position and the peak powder concentration is always located near the nozzle exit at a distance of 5 mm for the proposed nozzle design. Using a nozzle exit at the outward position, the powder stream can form a clear focused stream. The powder concentration tends to reduce with increasing values of the gas ¯ow velocities. The peak concentration of the stream focus is located at a distance ranging from 8 to 14 mm from the nozzle exit, but shifts towards the nozzle exit with increasing gas ¯ow velocities. The powder concentration in a nozzle at the inward position tends to reduce the peak powder concentration by up to 50% of that at the outward position.
[1] J.L. Koch, J. Mazumder, Rapid prototyping by laser cladding, in: ICALEO, 1993, pp. 556±559. [2] J. Lin, W.M. Steen, Design characteristics and development of a nozzle for coaxial laser cladding, J. Laser Appl. 10 (1998) 55±63. [3] P.-A. Vetter, Th. Engel, J. Fontaine, Laser cladding: the relative parameters for process control, SPIE Proc. 2207 (1994) 452±462. [4] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1979. [5] G.N. Abramovich, The Theory of Turbulent Jets, MIT Press, Cambridge, MA, 1963. [6] J.O. Hinze, Turbulence, McGraw-Hill, New York, 1975. [7] N. Rajaratnam, Turbulent Jets, Elsevier, Amsterdam, 1976. [8] R.A. Antonia, R.W. Bilger, An experimental investigation of an axisymmetric jet in a co-¯owing air stream, J. Fluid Mech. 61 (1973) 805±822. [9] J. Fan, H. Zhao, K. Chen, An experimental study of two-phase turbulent coaxial jets, Experiments in Fluids 13 (1992) 279±287. [10] V.L. Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1961. [11] FLUENT 4.4.4 User Guide, Fluent Inc., 1996. [12] C.T. Crowe, Review Ð numerical models for dilute gas±particle ¯ows, ASME Trans. J. Fluids Eng. 104 (1982) 297±303. [13] M.P. Sharma, C.T. Crowe, A novel physico-computational model for quasi one-dimensional gas±particle ¯ows, ASME Trans. J. Fluid Eng. 100 (1978) 343±349.
Acknowledgements Many thanks are due to Professor W.M. Steen of the University of Liverpool for his invaluable suggestions in this work, which is sponsored by NSC, Taiwan.