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´

• pez

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

1

Introduction

2

Model 1: constant normal velocity The model Numerical results

3

Model 2: linear velocity The model Numerical simulations and analysis

4

Model 3: exponential velocity

5

Conclusions

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. SiO2 + 4HF − → SiF4 + 2H2O PbO + H2SO4 − → PbSO4 + H2O K2O + 2HF − → 2KF + H2O SiF4 + 2HF − → H2SiF6 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 Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Model 1: constant normal velocity

s(x, t) surface. F(x, z, t) = z − s(x, t) = 0. n =

∇F ∇F = (−sx,1)

√

1+s2

x .

Material Derivative

∂F ∂t + vn ∇F = 0

vn rate removal surface First Model Equation st = −v

• 1 + s2

x

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Model 1: constant normal velocity

s(x, t) surface. F(x, z, t) = z − s(x, t) = 0. n =

∇F ∇F = (−sx,1)

√

1+s2

x .

Material Derivative

∂F ∂t + vn ∇F = 0

vn rate removal surface First Model Equation st = −v

• 1 + s2

x

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Charpit Method

Non-dimensionalisated equation st = −

• 1 + s2

x

F(x, t, s, p, q) = q +

• 1 + p2 = 0,

p = sx, q = st Problem            ˙ x = Fp ˙ t = Fq ˙ s = pFp + qFq ˙ p = −Fx − pFs ˙ q = −Ft − qFs

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Problem and Solution

               ˙ x =

p

√

1+p2

˙ t = 1 ˙ s = −

1

√

1+p2

˙ p = 0 ˙ q = 0 S(X(ξ, t), t) = S0(ξ) −

1

√

1+S′2

t            x = ξ t = 0 s = S0(ξ) = A sin(ξ) p = S′

0(ξ)

q = −

• (1 + S′

0(ξ)2)

X(ξ, t) = ξ +

S′

√

1+S′2

t.

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Plotting with Matlab

t ∈ [0, 3] x ∈ [−15, 15]

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Plotting with Matlab

t ∈ [0, 3] , x ∈ [0, 2π] time step= 1, space step = 1

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical results

Plotting with COMSOL Multiphysics

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

Model 2: linear velocity

Linear relationship between velocity and surface curvature k. v = v0 + v1κ. κ = −

sxx (1+s2

x )3/2 .

Second Model Equation st = −v0

• 1 + s2

x + v1

sxx 1 + s2

x

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

Model 2: linear velocity

Linear relationship between velocity and surface curvature k. v = v0 + v1κ. κ = −

sxx (1+s2

x )3/2 .

Second Model Equation st = −v0

• 1 + s2

x + v1

sxx 1 + s2

x

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

Numerical simulations

• 1. Non-dimensionalization

st = −

• 1 + s2

x

1

2 + ǫ

sxx 1 + s2

x

ǫ = v1 l v0

• 2. Finite elements method (COMSOL).

(video) Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

Critical ǫ value

ǫ values Results 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 Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

Finite Difference Method (Matlab)

Numerical discretisation: St = Sn+1 − Sn τ Sx = Sn(x + h) − Sn(x − h) 2h Sxx = Sn+1(x − h) − 2Sn+1(x) + Sn+1(x + h) h2

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

Solutions

Critical ǫ value.

Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions The model Numerical simulations and analysis

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. ⇓ v = v0 + v1k = v0

• 1 + v1

v0 k

• ∼

= v0 exp

• −v1sxx

v0(1+s2

x ) 3 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: st = −v

• 1 + s2

x

• 2. t∗ = l

v t∗ c (A)

• 3. Model 2, v linearly dependent on k (v = v0 + v1k). Diffusion

equation: st = −

• 1 + s2

x

1

2 + ǫ

sxx 1 + s2

x

• 4. ǫ = v1

l v0 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

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions

Bibliography

Fritz John, Partial differential equations p21, Springer-Verlag New York Inc. ISBN 0387906096 Lawrence C. Evans, Partial differential equations, chapter 3.2, American Mathematical Society. ISBN 0821807722 Aris and Amundson, First Order Partial Differential equations with Applications Volume 2 p49, Prentice

• Hall. ISBN 0135610923
• P. Ferster, S.C. Muller, B. Hesso, Critical size and curvature of wave formation in an excitable chemical

medium, Proc. Nail. Acad. Sci. USA vol 86 p 6831-6834, September 1989 Web of knowledge→Web of Science http://www.accesowok.fecyt.es/ Google http://www.google.es/ Wikipedia http://en.wikipedia.org/wiki/Main Page MACSI (Mathematics applications Consortium for Science and Industry) http://www.macsi.ie Infante, Juan Antonio, Numerical Methods Notes Polishing Lead Crystal Glass

Introduction Model 1: constant normal velocity Model 2: linear velocity Model 3: exponential velocity Conclusions

...questions?

Polishing Lead Crystal Glass