Polishing Lead Crystal Glass Universit` a degli Studi di Firenze - PowerPoint PPT Presentation

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Polishing Lead Crystal Glass Universit` a degli Studi di Firenze University of Oxford Universidad Complutense de Madrid June 24, 2008 Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The Team Agnese Bondi Francisco L´ opez Luca Meacci Cristina P´ erez Luis Felipe Rivero Elena Romero Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Summary Introduction 1 Model 1: constant normal velocity 2 The model Numerical results Model 2: linear velocity 3 The model Numerical simulations and analysis Model 3: exponential velocity 4 Conclusions 5 Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Introduction Irish manifacturer produces lead crystal glasses. They become opaque and rough after the cutting process. Polishing with immersion in acid. Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Polishing process Acid immersion → Rinsing process → Settle down Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Polishing process Acid immersion → Rinsing process → Settle down Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Polishing process Acid immersion → Rinsing process → Settle down Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Polishing process Acid immersion → Rinsing process → Settle down Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Polishing process Reactions. SiO 2 + 4 HF − → SiF 4 + 2 H 2 O PbO + H 2 SO 4 − → PbSO 4 + H 2 O K 2 O + 2 HF − → 2 KF + H 2 O SiF 4 + 2 HF − → H 2 SiF 6 Oxid + Acid = Salts. Soluble salts disappear in the water. Insoluble salts precipitate. Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions What is the problem? How does the process work? How long should the glass be immersed? Optimising the problem? Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions General assumptions One dimensional problem Initial form as the roughness: sinus . Homogeneous Neumann conditions. Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Model 1: constant normal velocity s ( x , t ) surface. F ( x , z , t ) = z − s ( x , t ) = 0. �∇ F � = ( − s x , 1) ∇ F √ n = x . 1+ s 2 Material Derivative ∂ F ∂ t + v n �∇ F � = 0 v n rate removal surface First Model Equation � 1 + s 2 s t = − v x Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Model 1: constant normal velocity s ( x , t ) surface. F ( x , z , t ) = z − s ( x , t ) = 0. �∇ F � = ( − s x , 1) ∇ F √ n = x . 1+ s 2 Material Derivative ∂ F ∂ t + v n �∇ F � = 0 v n rate removal surface First Model Equation � 1 + s 2 s t = − v x Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Charpit Method Non-dimensionalisated equation � 1 + s 2 s t = − x 1 + p 2 = 0 , � F ( x , t , s , p , q ) = q + p = s x , q = s t Problem  x = F p ˙  ˙  t = F q    s = pF p + qF q ˙ p = − F x − pF s ˙     q = − F t − qF s ˙  Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Problem and Solution p  √ x = ˙  x = ξ  1+ p 2     t = 0  ˙ t = 1       1 √ s = S 0 ( ξ ) = A sin( ξ ) s = − ˙ 1+ p 2 p = S ′ 0 ( ξ )     p = 0 ˙     � (1 + S ′ 0 ( ξ ) 2 )  q = −   q = 0 ˙  S ′ √ X ( ξ, t ) = ξ + 0 t . 1 √ S ( X ( ξ, t ) , t ) = S 0 ( ξ ) − t 1+ S ′ 2 1+ S ′ 2 0 0 Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Plotting with Matlab t ∈ [0 , 3] x ∈ [ − 15 , 15] Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Plotting with Matlab t ∈ [0 , 3] , x ∈ [0 , 2 π ] time step = 1, space step = 1 Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical results Model 3: exponential velocity Conclusions Plotting with COMSOL Multiphysics Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions Model 2: linear velocity Linear relationship between velocity and surface curvature k . v = v 0 + v 1 κ . s xx κ = − x ) 3 / 2 . (1+ s 2 Second Model Equation s xx � 1 + s 2 s t = − v 0 x + v 1 1 + s 2 x Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions Model 2: linear velocity Linear relationship between velocity and surface curvature k . v = v 0 + v 1 κ . s xx κ = − x ) 3 / 2 . (1+ s 2 Second Model Equation s xx � 1 + s 2 s t = − v 0 x + v 1 1 + s 2 x Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions Numerical simulations 1. Non-dimensionalization s xx � 1 2 + ǫ 1 + s 2 � s t = − x 1 + s 2 x ǫ = v 1 l v 0 2. Finite elements method (COMSOL). (video) Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions Critical ǫ value ǫ values Results 0 Previous model. ǫ > 0 . 141 The surface goes up at the beginning. 0 < ǫ ≤ 0 . 141 The surface always goes down. (video) Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions Finite Difference Method (Matlab) Numerical discretisation: S t = S n +1 − S n τ S x = S n ( x + h ) − S n ( x − h ) 2 h S xx = S n +1 ( x − h ) − 2 S n +1 ( x ) + S n +1 ( x + h ) h 2 Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions Solutions Critical ǫ value. Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity The model Model 2: linear velocity Numerical simulations and analysis Model 3: exponential velocity Conclusions About initial conditions A = a l where a = height and l = length . A = 0 . 5 A = 1 The bigger A is, the slower velocity goes. Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Model 3: exponential velocity Exponential relationship between normal velocity and surface curvature. ⇓ � � � � ∼ 1 + v 1 − v 1 s xx v = v 0 + v 1 k = v 0 = v 0 exp v 0 k 3 v 0 (1+ s 2 x ) 2 Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions Conclusions 1. Model 1, v as a constant. Hammilton-Jacobi non-linear equation: � 1 + s 2 s t = − v x 2. t ∗ = l v t ∗ c ( A ) 3. Model 2, v linearly dependent on k ( v = v 0 + v 1 k ). Diffusion equation: s xx � 1 2 + ǫ 1 + s 2 � s t = − x 1 + s 2 x 4. ǫ = v 1 l v 0 critical value, if it is too large it becomes unphysical. 5. Next step: Exponential problem. 6. Not as easy finding a proper velocity rate when several acids appear. New research? Polishing Lead Crystal Glass