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Scaling of the Cathode Region of a Long GTA Welding Arc
P. F. Mendez, M. A. Ramirez G. Trápaga, T. W. Eagar Massachusetts Institute of Technology August 23, 2000 I want to share with you some aspects of our ongoing work on modeling at the W&J group at MIT. In particular, a technique we are developing for which we see much promise as an auxiliary tool for the analysis of numerical models. I’ll use the modeling of the cathode region of the arc as an example.
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Motivation The arc is an essential component of a math model of the welding process The generalization of the numerical results is desirable for the interpretation and transmission of the findings These two seemingly different motivations will be addressed simultaneously. Emphasis will be put on generalization instead of details of the numerical solutions Weld properties -->heat transfer--> heat source
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Evolution of Arc Modeling
These are only a few selected references relative to arc modeling As we see, there’s no shortage of knowledge about the arc. The geometric complexity didn’t increase but the description of the physics involves many parameters. Squire 1951: point force, point heat source Shercliff:1969: point current source Maecker 1955: cathode spot Ramakrishnan 1978, Glickstein 1979: 1D models of heat and mass transfer Hsu 1983: 2D model McKelliget: 2D, anode conditions Choo 1990, Kim 1997: 2D, deformed free surface Lee 1996: 2D cathode tip angle As the complexity of the model increases, the characterization of the model requires exponentially more points
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Evolution of Arc Modeling
Modeling restrictions have been removed gradually The increase in complexity focused more on the physics than on the geometry Not all the parameters that describe the problem are simultaneously relevant The problem can be divided in different regimes in which different sets of parameters balance each other In each regime the description of the problem is very simple Property of engineering systems: can be divided in regimes with a particular balance in each. E.g. pressure can be balanced by inertial forces or viscous forces We developed a special technique for obtaining this balance of forces
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Order of Magnitude Scaling (OMS)
Determines dominant and balancing forces (without solving the PDE’s) Determines the range of validity of a particular balance Provides order of magnitude estimations of the solutions Ranks the relative importance of the dimensionless groups Permits the construction of universal maps of a process Uses dimensional analysis to obtain generality Is similar to the perturbation method of applied math. Important difference: No need to solve PDE’s
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Formulation of the problem
Cathode region: e.m. forces generate pressure pressure generates momentum no interaction with anode region little temperature variations From the complete numerical model we can extract the temperature profile
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Temperature variations in the cathode region
We see that the temperature variations within the cathode region are small compared to the variation elsewhere in the arc Hsu, 1983 Ramirez, 1999 200 A, 10 mm, Argon
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Governing Equations Unknown functions: continuity Navier-Stokes
Maxwell Unknown functions: Point out the different terms in the equations
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Problem Parameters The parameters that completely describe the problem as formulated are: r : density of the plasma (at max. temperature) m : viscosity of the plasma (at max. temperature) m0 : permeability of vacuum Rc : cathode radius Jc : cathode density Ra : anode radius h : arc length Rc can be deduced from the welding current. All the parameters are controllable experimentally except Ra.
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Boundary Conditions For the numerical model: I
The numerical model includes the anode model used by McKelliget J
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Boundary Conditions For Order of Magnitude Scaling:
Important: we can identify that the velocities have characteristic values, even if we don’t know what those values are. unknown characteristic values
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Behavior of the solutions
BC for order of OMS Real values (200 A, 10 mm) VZ VZ E H 40 256 F G 7 85 VZ 40 7 The functions show characteristic values The variation between characteristic values is smooth E F 256 VZ 85 H G
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Scaling functions For the electromagnetic field
unknown characteristic value The expressions DO NOT assume a parabolic profile. They assume a parabolic behavior only at the axis, which can be demonstrated analytically.
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Scaling functions For the fluid flow unknown characteristic values
The expressions DO NOT assume a parabolic profile. They assume a parabolic behavior only at the axis, which can be demonstrated analytically.
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Modified Dominant Balance
unknown functions vary smoothly normalization is performed The dominant and balancing forces can be determined without solving the differential equation
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Modified Dominant Balance
Radial inertial forces Axial inertial forces Balancing Radial pressure variation Radial e.m. forces Axial pressure variation
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Estimations of the characteristic parameters
Based on the balance obtained: Power-law expressions Through dimensional analysis is possible to demonstrate that the expressions obtained will always be of the form of a power law
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Estimations of the characteristic parameters
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Dimensional Analysis The dimensionless groups that govern the model are(show both natural and imposed dim groups)
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Range of Validity Since the coefficients in the normalized equations are the ratio of secondary forces to the dominant, the boundary between limiting cases can be defined when they are 1 Re h/Rc 1 102 10 Typical welding arc Since the coefficients depend on the dimensionless groups, a map can be constructed
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Corrections for the Estimations
The differences between the estimations and numerical calculations depend only on the governing dimensionless groups Considering only the more relevant dimensionless groups usually is accurate enough This is similar to perturbation analysis
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Corrections for the Estimations
Power laws are convenient expressions The small exponents indicate that the estimations capture most of the behavior of the model
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Corrections for the Estimations
correction function actual difference
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Universal Process Maps
Characteristic values are scaled and corrected universal maps of the process can be build based only on the problem parameters no need for empirical measurements (e.g. to get maximum velocity from numerical model or experiment) Put them on the internet
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Universal Process Maps
VR(R,Z)/VRS 200 A 10 mm 2160 A 70 mm Put them on the internet
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Discussion The estimations obtained are comparable to those available in the literature (Maecker 1955) Accuracy of the estimations can be increased by using numerical results or experiments relevant dimensionless groups The effect of simplifications (e.g. constant properties appear as error in the correction functions) The contour maps suggest ways of improving the numerical model In sharper electrodes the e.m. forces also generate momentum without increase in pressure application to GMAW? one dimensionless group (Ra/Rc) could not be estimated with this isothermal model
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Discussion Is there a unique solution to the dominant balance?
Can dominant balance be fooled? Does normalization solve potential paradoxes in dominant balance? one dimensionless group (Ra/Rc) could not be estimated with this isothermal model
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Conclusions Viscous effects are small
Electromagnetic forces create pressure, which is balanced by inertial forces thermal expansion is secondary Power-law form equations, based only on the parameters of the problem provide: properties of the fluid flow correction functions Universal maps of the arc can be generated they can be scaled to a wide range of arcs
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State of the Art in Arc Modeling
Tekriwal, Mazunder D analytical model for heat source (pointed tip). Convection at anode by means of heat transfer coefficient, properties constant Morrow & Lowke D theory for the electric sheats of electric arcs. (anode and cathode falls). Zhu & Lowke Treats cathode boundary layer and arc column as unified system. Mckelliget & Szekely.1986 cathode and anode development Delalondre & Simonin D. modeling high intensity arcs inclu- ding non equilibrium description of the cathode sheath. Auttukhov, 1983 Current thermionic emission Cram, Focussing on the energy balance of the electrode Hsu & Pfender,1983. Detailed model for the cathode region Chen, David, Zacharia, 1997 model interaction between the and the weld pool in GTA. Free surface changing with time Kim, Fan, Na D GTA, cathode influence and free surface. No assumption on Jc. Choo, Couple between arc and weld pool. Deformation of the pool. Westhoff, D arc model specifying current density at anode. deformation of weld pool. Small changes in Jc changes T fields. Chen & Zacharia, 1991. Analysis of the electrode tip angle and geometry of the GTA weld pool. Sui & Kou, D, Effect of the tip geometry shielding gas, nozzle, Dawson, Bendzak, Mueller, 1997 Fluid flow and Heat Transfer in a twin cathode DC furnace. Exp. messurments in lab. modeling furnace Qian, Farouk, Mathasaran, 1995 Fluid flow and heat transfer in EAF, 2D, DC McKelliget & Szekely, 1983 2D arc model, coupling pool and arc models. McKelliget & Swzekely, 1981. Heat transfer & Fluid flow in a EAF Dinulesca & Pfender, 1980 Analysis of the boundary layer in high intensity arcs Ushio, Szekely, Chang, 1980 2D model, assuming parabolic current density distribution, k-e turbulent model. Pradip, Yogadra, Rama, 1995 3D, heat transfer, fluid flow in GTAW with non-axisymetric b.c.’s. Maxwell equations, uses buoyancy, surface tension and electromagnetic forces. McKelliget & Szekely, D, DC approximate boundary conditions. Jc=65MA/m 2 Kovitya & Cram, D, LTE MHD, boundary conditions assumed. Kovitya & Lowke, D, uses properties theoretically calculated. Jc=100MA/m . Extends Lowke’s ,odel to incorporate Lorentz’s for- ces and electron drift enthalpy. Hsu, Etemadi, Pfender, D MHD eqs. with b.c’s experimentally determined. Anode and cathode excluded. Jc assumed with a gaussian shape. Allum, D Assumes current and velocity in gaussian profiles. Includes magnetic, viscous and gravitational forces. Ramakrishan & Nou 2D, semianalytical model. Radial vel. field assumed. Lowke, D, analytical model for arc voltage, electric field and plasma velocity. Chang, Eagar, Szekely Velo- city fields calculated analytically using Lorentz’s forces. Goldak & Moore Finite ele- ment method. Describes the source. Glickstein, D, analytical. Radial variations of temperature and J. No plas- ma flow. Lowke D continuity energy, naturla convection. lowke, d momentum and en ergy eqs. Natural convection. Low currents. 3 1 2A 2B 2AA 2AB 2AC 2C ARC COLUMN MODELING BASED ON PHYSICAL PRINC. B.C.’S APPROXIMATED OR EXP. DETERMINED. EAF WELDING RF DISCHARGES ELECTRODE BEHAVIOR CATHODE AND ANODE FALLS Squire 51: isothermal, point force Maecker 55: e.m. force approx Shercliff 69: point current Yas’ko 69: dimensional analysis 1=RF Discharges (nat. convection). 2=Welding (Laminar flow). 3=EAF (turbulent flow). 2A=B.C.(anode and cathode modeling) 2B=Coupled arc and weld pool (welding) 2C=Geometry effects in welding. 2AA=ANODE REGION 2AB= ANODE AND CATHODE 2AC=CATHODE REGION As we see, there’s no shortage of knowledge about the arc. Modeling restrictions have been removed gradually. Let’s see how these models evolved in complexity
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