Order of Magnitude Scaling Technique I developed at MIT to deal with complex materials processing problems, and seems to be very general. Patricio F. Mendez Stuttgart, November 5, 2001

Design of an Electric Arc Furnace Scaling laws? Dominant forces? Secondary forces? Regimes? Start with example of Electric Furnace I earned a Mechanical Engineering degree at the UBA, and worked for two years at Techint steelmaking plant in Argentina before coming to MIT for my Ph.D. At MIT, I was involved in many activities, from IM soccer to being an entrepreneur, where I co-invented a machine that’s used to make parts for the Ford Focus.

Evolution in Complexity Modeling of the Plasma Arc The complexity of the systems considered jumped with the introduction of personal computers, and it is increasing rapidly The tools of analysis we have are good for the old problems but not for the new problems

Challenges Reality Equations Result Application Solving the governing equations Writing the appropriate equations Obtaining empirical evidence Expanding scope into larger systems

Novel Technique: OMS Applied Mathematics Engineering Dominant balance, asymptotics Impractical for systems of equations Engineering Dimensional analysis, inspectional analysis, scaling Little help when many parameters Artificial Intelligence Automated reasoning, order of magnitude reasoning In early stages for differential equations Two stages for OMS matrix of coefficients: human input search of self consistent solution: automatic I’ll demonstrate it with an example of the plasma arc, although this technique is good for a wide range of problems in heat transfer, fluid flow, electromagnetism, and could be applied to fields outside engineering such as financial engineering

Basic Knowledge of Behavior Temperature varies smoothly in each region Cathode region Column Anode region Two things necessary to build matrix of coefficients: basic knowledge of behavior of system properties of engineering equations Gas Plasma Gas

Governing Equations Normalization Parameters: P Energy in plasma Normalization Energy in gas Parameters: P “Interface” plasma-gas Equations in engineering have the form of a sum of terms Each term has a coefficient with the form of a power law and a function Power law involves parameters and characteristic values unknown scaling factor: S Maxwell OM(1) Coefficient: C

Matrix of Coefficients Parameters Unknowns coefficients parameters unknowns Coefficient

Mathematical Operations Normalization of equations subtract rows dominant term=1 Asymptotic regimes: balancing term= dominant term

Dominant Balance Radiation Electron drift Joule R Joule Z Conduction cooling heating inconsistent incompatible self-consistent

Approximate Solutions parameters exponents unknowns

Approximate Solutions 1 day+0.001 s/point R2=92% Other Applications in Engineering Heat and fluid flow in welding Non-Isothermal boundary layer Ceramic to metal bonding Polymer fiber retraction 1 PhD+2 h/point

Heat and Fluid Flow in Welding What causes this crater? Equations continuity Navier Stokes energy Maxwell free surface Driving forces gas shear arc pressure Marangoni capillary electromagnetic gravity buoyancy 9 unknowns, non linear PDE’s 7 driving forces

Heat and Fluid Flow in Welding arc pressure / viscous electromagnetic / viscous hydrostatic / viscous capillary / viscous Marangoni / gas shear buoyancy / viscous gas shear / viscous convection / conduction inertial / viscous diff.=/diff.^ gas shear causes crater Mention this is one of the questions we had at the beginning Show secondary forces that can improve accuracy: Natural dimensionless groups Mention the Holy Grail of dimensional analysis

Non-Isothermal Boundary Layer What are the regimes? Equations continuity Navier Stokes energy Driving forces conduction convection

Non-Isothermal Boundary Layer 666 iterations Mention applications outside engineering Geology, Astrophysics Biology Financial engineering Teaching

Conclusions Closed-form approximate solutions Natural dimensionless groups Regimes Proven for engineering applications Matrix of coefficients: Formalism for further research Potential beyond engineering

Modeling of the Plasma Arc Scope of OMS Year of publication Modeling of the Plasma Arc

OMS: Related Techniques inspectional analysis Szirtes 1998, Chen 1971, Barr 1987 Differential Equations Dimensional Analysis Matrix Algebra similarity dominant balance Bender and Orszag, 1978 Chen, 1990; Barenblatt, 1996 intermediate asymptotics AOM, Yip, 1996 characteristic values Denn, 1980 Asymptotic Considerations Order of Magnitude Reasoning

Approximate Solutions