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Modelling decontamination of two-dimensional spills Oliver - - PowerPoint PPT Presentation

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University of Oxford Modelling decontamination of two-dimensional spills Oliver Whitehead Mathematical Institute Supervisors: Chris Breward, Ian Griffiths (Oxford), Ross Heatlie- Branson (DEFRA) and with Ellen Luckins EPSRC RC Centre for

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SLIDE 1

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Modelling decontamination of two-dimensional spills

Oliver Whitehead Supervisors: Chris Breward, Ian Griffiths (Oxford), Ross Heatlie- Branson (DEFRA) and with Ellen Luckins

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SLIDE 2

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Non-porous Medium (Air) Porous Medium, (Concrete)

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SLIDE 3

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Non-porous Medium (Air) Porous Medium, (Concrete)

Agent

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SLIDE 4

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Non-porous Medium (Air) Porous Medium, (Concrete)

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SLIDE 5

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Non-porous Medium (Air) Porous Medium, (Concrete)

‘The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone

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SLIDE 6

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Porous Medium, (Concrete) Non-porous Medium (Air)

‘The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone

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SLIDE 7

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Porous Medium, (Concrete) Non-porous Medium (Air)

Cleanser ‘The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone

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SLIDE 8

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Porous Medium, (Concrete) Non-porous Medium (Air)

Cleanser ‘The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone

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SLIDE 9

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Modelling Assumptions

  • Agent and cleanser are immiscible
  • The agent is neat – not in solution
  • The reaction occurs only at the interface and is:
  • Agent (𝑚) + Cleanser (𝑏𝑟) Product
  • Product insoluble in the agent
  • Product in solution does not affect cleanser in solution
  • Assume that there is no fluid flow
  • Below the agent is unsaturated
  • No evaporation of cleanser or agent
  • The layers of cleanser and agent are thin
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SLIDE 10

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Porous Medium, (Concrete) Non-porous Medium (Air)

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SLIDE 11

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Porous Medium, (Concrete) Non-porous Medium (Air) Height Length

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SLIDE 12

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up

Porous Medium, (Concrete) Non-porous Medium (Air) Height

𝐼𝑓𝑗𝑕ℎ𝑢 𝑀𝑓𝑜𝑕𝑢ℎ ≈ 0.1 ≪ 1

Length

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SLIDE 13

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

𝑍 𝑌

Mathematical Model

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SLIDE 14

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A

𝑍 𝑌

Mathematical Model

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

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SLIDE 15

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B

𝑍 𝑌

Mathematical Model

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

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SLIDE 16

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B

𝑍 𝑌

C

Mathematical Model

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent

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SLIDE 17

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B D

𝑍 𝑌

C

Mathematical Model

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium

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SLIDE 18

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 D

𝑍 = ℎ(𝑌) 𝑍 𝑌

C

Mathematical Model

𝒐𝒖𝒑𝒒

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0

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SLIDE 19

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 2 D

𝑍 = ℎ(𝑌) 𝑍 𝑌

C

Mathematical Model

𝒐𝒖𝒑𝒒

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = ϕ𝐷, 𝑑𝑍 = ϕ𝐷𝑍

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SLIDE 20

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 2 D

𝑍 = ℎ(𝑌) 𝑍 𝑌

C

Mathematical Model

3 𝒐𝒖𝒑𝒒

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = ϕ𝐷, 𝑑𝑍 = ϕ𝐷𝑍 3. No flux, 𝐷𝑍 = 0

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SLIDE 21

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 2 D 4

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

C

Mathematical Model

𝒐𝑰 3 𝒐𝒖𝒑𝒒

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = ϕ𝐷, 𝑑𝑍 = ϕ𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + 𝐸𝑑(𝒐𝑰 · 𝛼)𝐷 = −𝑙𝐷, 𝜖𝐼 𝜖𝑢 = 𝑙χ𝐷

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SLIDE 22

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 2 D 4

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

C

Mathematical Model

𝒐𝑰 3 𝒐𝒄𝒑𝒖 𝒐𝒖𝒑𝒒 5

𝑍 = −𝑒(𝑌) A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = ϕ𝐷, 𝑑𝑍 = ϕ𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + 𝐸𝑑(𝒐𝑰 · 𝛼)𝐷 = −𝑙𝐷, 𝜖𝐼 𝜖𝑢 = 𝑙χ𝐷

5. No flux, 𝒐𝒄𝒑𝒖 · 𝛼𝑑 = 0

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SLIDE 23

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 2 D 4

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

C

Mathematical Model

𝒐𝑰 3 𝒐𝒄𝒑𝒖 𝒐𝒖𝒑𝒒 5

𝑍 = −𝑒(𝑌) Initial conditions: 𝑑 = 𝑑∗ 𝐼 = 0 A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = ϕ𝐷, 𝑑𝑍 = ϕ𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + 𝐸𝑑(𝒐𝑰 · 𝛼)𝐷 = −𝑙𝐷, 𝜖𝐼 𝜖𝑢 = 𝑙χ𝐷

5. No flux, 𝒐𝒄𝒑𝒖 · 𝛼𝑑 = 0

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SLIDE 24

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

  • ral

al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

A B 1 2 D 4

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

C

Mathematical Model

A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝐸𝑑𝛼2𝑑

B.

𝜖𝐷 𝜖𝑢 = 𝐸𝑑𝛼2𝐷

C. Neat Agent D. Unsaturated medium

𝒐𝑰

Non-dimensionalise: 𝑑~𝑑∗ 𝐷~𝑑∗/ϕ 𝑌~𝑀 ℎ, 𝐼, 𝑍~ℎ∗ 𝑢~ℎ∗2/𝐸𝑑

3 𝒐𝒄𝒑𝒖 𝒐𝒖𝒑𝒒 5

𝑍 = −𝑒(𝑌) 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = ϕ𝐷, 𝑑𝑍 = ϕ𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + 𝐸𝑑(𝒐𝑰 · 𝛼)𝐷 = −𝑙𝐷, 𝜖𝐼 𝜖𝑢 = 𝑙χ𝐷

5. No flux, 𝒐𝒄𝒑𝒖 · 𝛼𝑑 = 0 Initial conditions: 𝑑 = 𝑑∗ 𝐼 = 0

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SLIDE 25

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Dimensionless Model

A B 1 2 3 D 4 C 𝒐𝒖𝒑𝒒

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

5 𝒐𝑰 𝒐𝒄𝒑𝒖

𝑍 = −𝑒(𝑌) A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝜁2 𝜖2𝑑 𝜖𝑌2 + 𝜖2𝑑 𝜖𝑍2

B.

𝜖𝐷 𝜖𝑢 = 𝜁2 𝜖2𝐷 𝜖𝑌2 + 𝜖2𝐷 𝜖𝑍2

C. Neat Agent D. Unsaturated medium Non-dimensionalise: 𝑑~𝑑∗ 𝐷~𝑑∗/ϕ 𝑌~𝑀 ℎ, 𝐼, 𝑍~ℎ∗ 𝑢~ℎ∗2/𝐸𝑑 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = 𝐷, 𝑑𝑍 = 𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + (𝒐𝑰 · 𝛼)𝐷 = −𝐿𝐷, 𝜖𝐼 𝜖𝑢 = 𝐿𝜉𝐷

5. No flux, 𝒐𝒄𝒑𝒖 · 𝛼𝑑 = 0 Initial conditions: 𝑑 = 1 𝐼 = 0

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SLIDE 26

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Dimensionless Model

A B 1 2 3 D 4 C 𝒐𝒖𝒑𝒒

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

5 𝒐𝑰 𝒐𝒄𝒑𝒖

𝑍 = −𝑒(𝑌) A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝜁2 𝜖2𝑑 𝜖𝑌2 + 𝜖2𝑑 𝜖𝑍2

B.

𝜖𝐷 𝜖𝑢 = 𝜁2 𝜖2𝐷 𝜖𝑌2 + 𝜖2𝐷 𝜖𝑍2

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = 𝐷, 𝑑𝑍 = 𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + (𝒐𝑰 · 𝛼)𝐷 = −𝐿𝐷, 𝜖𝐼 𝜖𝑢 = 𝐿𝜉𝐷

5. No flux, 𝒐𝒄𝒑𝒖 · 𝛼𝑑 = 0 Initial conditions: 𝑑 = 1 𝐼 = 0

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SLIDE 27

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Dimensionless Model

A B 1 2 3 D 4 C 𝒐𝒖𝒑𝒒

𝑍 = ℎ(𝑌) 𝑍 = −𝐼(𝑌, 𝑢) 𝑍 𝑌

5 𝒐𝑰 𝒐𝒄𝒑𝒖

𝑍 = −𝑒(𝑌) A. Diffusion of cleanser,

𝜖𝑑 𝜖𝑢 = 𝜁2 𝜖2𝑑 𝜖𝑌2 + 𝜖2𝑑 𝜖𝑍2

B.

𝜖𝐷 𝜖𝑢 = 𝜁2 𝜖2𝐷 𝜖𝑌2 + 𝜖2𝐷 𝜖𝑍2

C. Neat Agent D. Unsaturated medium 1. No flux, 𝒐𝒖𝒑𝒒 · 𝛼𝑑 = 0 2. Continuity, 𝑑 = 𝐷, 𝑑𝑍 = 𝐷𝑍 3. No flux, 𝐷𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼

𝜖𝑢 + (𝒐𝑰 · 𝛼)𝐷 = −𝐿𝐷, 𝜖𝐼 𝜖𝑢 = 𝐿𝜉𝐷

5. No flux, 𝒐𝒄𝒑𝒖 · 𝛼𝑑 = 0 Initial conditions: 𝑑 = 1 𝐼 = 0 Dimensionless groupings: ε = ℎ∗ 𝑀 𝐿 = 𝑙ℎ∗ 𝐸𝑑 𝜉 = χ𝑑∗ ϕ

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SLIDE 28

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

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SLIDE 29

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

We find three main timescales over which effects occur:

slide-30
SLIDE 30

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

  • ral

al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

We find three main timescales over which effects occur:

  • Timescale for the chemical reaction
slide-31
SLIDE 31

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

  • ral

al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

We find three main timescales over which effects occur:

  • Timescale for the chemical reaction
  • Timescale for vertical diffusion
slide-32
SLIDE 32

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

  • ral

al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

We find three main timescales over which effects occur:

  • Timescale for the chemical reaction
  • Timescale for vertical diffusion
  • Timescale for horizontal diffusion
slide-33
SLIDE 33

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

≈4 minutes ≈ 7 hours We find three main timescales over which effects occur:

  • Timescale for the chemical reaction
  • Timescale for vertical diffusion
  • Timescale for horizontal diffusion
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SLIDE 34

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

𝜁2 𝜁 1 𝜁−1 𝐿

≈4 minutes ≈ 7 hours We find three main timescales over which effects occur:

  • Timescale for the chemical reaction
  • Timescale for vertical diffusion
  • Timescale for horizontal diffusion
slide-35
SLIDE 35

University of Oxford Mathematical Institute

EPSRC RC Centre for Doctor

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Timescale Analysis

𝜁2 𝜁 1 𝜁−1 𝐿

≈4 minutes ≈ 7 hours We find three main timescales over which effects occur:

  • Timescale for the chemical reaction
  • Timescale for vertical diffusion
  • Timescale for horizontal diffusion

We are interested in seeing what can be decontaminated over a short time decontamination.

slide-36
SLIDE 36

University of Oxford Mathematical Institute

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𝑳 ≫ 𝑷(𝟐)

𝜁2 𝜁 1 𝜻−𝟐 𝐿

slide-37
SLIDE 37

University of Oxford Mathematical Institute

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𝑳 ≫ 𝑷(𝟐)

  • Reaction rate faster than the vertical or horizontal diffusion

𝜁2 𝜁 1 𝜻−𝟐 𝐿

slide-38
SLIDE 38

University of Oxford Mathematical Institute

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al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

𝑳 ≫ 𝑷(𝟐)

  • Reaction rate faster than the vertical or horizontal diffusion
  • Clean-up is limited by the time it takes cleanser to diffuse towards

the boundary

𝜁2 𝜁 1 𝜻−𝟐 𝐿

slide-39
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𝑳 ≫ 𝑷(𝟐)

  • Reaction rate faster than the vertical or horizontal diffusion
  • Clean-up is limited by the time it takes cleanser to diffuse towards

the boundary

  • So, no matter how powerful you make the cleanser, the clean-up

time does not reduce considerably beyond this limit

𝜁2 𝜁 1 𝜻−𝟐 𝐿

slide-40
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𝑳 ≫ 𝑷(𝟐)

  • Reaction rate faster than the vertical or horizontal diffusion
  • Clean-up is limited by the time it takes cleanser to diffuse towards

the boundary

  • So, no matter how powerful you make the cleanser, the clean-up

time does not reduce considerably beyond this limit

  • Equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution C0 = 0,

𝜖𝐼0 𝜖𝑢 = −𝜉 𝜖𝐷0 𝜖𝑍

at reaction front

𝜁2 𝜁 1 𝜻−𝟐 𝐿

slide-41
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𝑳 ≫ 𝑷(𝟐)

  • Reaction rate faster than the vertical or horizontal diffusion
  • Clean-up is limited by the time it takes cleanser to diffuse towards

the boundary

  • So, no matter how powerful you make the cleanser, the clean-up

time does not reduce considerably beyond this limit

  • Equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution 𝐷0 = 0,

𝜖𝐼0 𝜖𝑢 = −𝜉 𝜖𝐷0 𝜖𝑍

at reaction front

  • So concentration does not change over the reaction timescale

except for in a boundary layer near Y = 0.

𝜁2 𝜁 1 𝜻−𝟐 𝐿

slide-42
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𝑳~𝑷(𝟐)

𝜁2 𝜁 1 𝜁−1 𝐿

slide-43
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𝑳~𝑷(𝟐)

  • If the reaction occurs on a similar timescale to vertical diffusion, the

cleanser at the boundary is not all reacted away immediately

𝜁2 𝜁 1 𝜁−1 𝐿

slide-44
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𝑳~𝑷(𝟐)

  • If the reaction occurs on a similar timescale to vertical diffusion, the

cleanser at the boundary is not all reacted away immediately

  • The horizontal diffusion timescale is still much slower

𝜁2 𝜁 1 𝜁−1 𝐿

slide-45
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𝑳~𝑷(𝟐)

  • If the reaction occurs on a similar timescale to vertical diffusion, the

cleanser at the boundary is not all reacted away immediately

  • The horizontal diffusion timescale is still much slower
  • So the cleanser reacts vertically into the agent

𝜁2 𝜁 1 𝜁−1 𝐿

slide-46
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𝑳~𝑷(𝟐)

  • If the reaction occurs on a similar timescale to vertical diffusion, the

cleanser at the boundary is not all reacted away immediately

  • The horizontal diffusion timescale is still much slower
  • So the cleanser reacts vertically into the agent

𝜁2 𝜁 1 𝜁−1 𝐿

  • The equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution 𝐷0

𝜖𝐼0 𝜖𝑢 + 𝜖𝐷0 𝜖𝑍 = −𝐿𝐷0, 𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝐷0

at reaction front

𝜖𝐷0 𝜖𝑍 = 0

at top of cleanser and bottom of agent

slide-47
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  • The equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution 𝐷0

𝜖𝐼0 𝜖𝑢 + 𝜖𝐷0 𝜖𝑍 = −𝐿𝐷0, 𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝐷0

at reaction front

𝜖𝐷0 𝜖𝑍 = 0

at top of cleanser and bottom of agent

𝑳~𝑷(𝟐)

𝜁2 𝜁 1 𝜁−1 𝐿

slide-48
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𝐿 = 1, 𝜉 = 1

𝑳~𝑷(𝟐)

𝜁2 𝜁 1 𝜁−1 𝐿

  • The equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution 𝐷0

𝜖𝐼0 𝜖𝑢 + 𝜖𝐷0 𝜖𝑍 = −𝐿𝐷0, 𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝐷0

at reaction front

𝜖𝐷0 𝜖𝑍 = 0

at top of cleanser and bottom of agent

slide-49
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𝐿 = 1, 𝜉 = 1

𝑳~𝑷(𝟐)

𝜁2 𝜁 1 𝜁−1 𝐿

  • The equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution 𝐷0

𝜖𝐼0 𝜖𝑢 + 𝜖𝐷0 𝜖𝑍 = −𝐿𝐷0, 𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝐷0

at reaction front

𝜖𝐷0 𝜖𝑍 = 0

at top of cleanser and bottom of agent 𝐿 = 1, 𝜉 = 4

slide-50
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𝐿 = 1, 𝜉 = 4

𝑳~𝑷(𝟐)

𝜁2 𝜁 1 𝜁−1 𝐿

  • The equations reduce to:

𝜖𝑑0 𝜖𝑢 = 𝜖2𝑑0 𝜖𝑍2, 𝜖𝐷0 𝜖𝑢 = 𝜖2𝐷0 𝜖𝑍2

in cleanser solution 𝐷0

𝜖𝐼0 𝜖𝑢 + 𝜖𝐷0 𝜖𝑍 = −𝐿𝐷0, 𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝐷0

at reaction front

𝜖𝐷0 𝜖𝑍 = 0

at top of cleanser and bottom of agent

slide-51
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Later time dynamics

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Later time dynamics

  • In each of these simulations, vertical reaction dominates the early

time dynamics

slide-53
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Later time dynamics

  • In each of these simulations, vertical reaction dominates the early

time dynamics

  • In later time, the interface can start to move laterally
slide-54
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Later time dynamics

  • In each of these simulations, vertical reaction dominates the early

time dynamics

  • In later time, the interface can start to move laterally

𝐿 = 1, 𝜉 = 4

slide-55
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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

𝜁2 𝜻 1 𝜁−1 𝐿

slide-56
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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

  • Reaction much slower than vertical diffusion timescale

𝜁2 𝜻 1 𝜁−1 𝐿

slide-57
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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

  • Reaction much slower than vertical diffusion timescale
  • So, vertical mixing occurs very fast

𝜁2 𝜻 1 𝜁−1 𝐿

slide-58
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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

  • Reaction much slower than vertical diffusion timescale
  • So, vertical mixing occurs very fast
  • Reaction also much faster than horizontal diffusion timescale

𝜁2 𝜻 1 𝜁−1 𝐿

slide-59
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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

  • Reaction much slower than vertical diffusion timescale
  • So, vertical mixing occurs very fast
  • Reaction also much faster than horizontal diffusion timescale
  • So, no lateral mixing occurs

𝜁2 𝜻 1 𝜁−1 𝐿

slide-60
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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

  • Reaction much slower than vertical diffusion timescale
  • So, vertical mixing occurs very fast
  • Reaction also much faster than horizontal diffusion timescale
  • So, no lateral mixing occurs

𝜁2 𝜻 1 𝜁−1 𝐿

  • The equations reduce to an implicit equation for 𝐼 and 𝐷:

𝐼0 + 𝜉 + 1 ℎ ln 1 −

𝐼0 𝜉ℎ + 𝑢 𝐿 = 0

𝐷0 =

ℎ−𝐼0/𝜉 ℎ+𝐼0

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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

𝜁2 𝜻 1 𝜁−1 𝐿

  • The equations reduce to an implicit equation for 𝐼 and 𝐷:

𝐼0 + 𝜉 + 1 ℎ ln 1 −

𝐼0 𝜉ℎ + 𝑢 𝐿 = 0

𝐷0 =

ℎ−𝐼0/𝜉 ℎ+𝐼0

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𝑷(𝜻𝟑) ≪ 𝑳 ≪ 𝑷(𝟐)

𝜁2 𝜻 1 𝜁−1 𝐿

𝐿 = 𝜁, 𝜉 = 4

  • The equations reduce to an implicit equation for 𝐼 and 𝐷:

𝐼0 + 𝜉 + 1 ℎ ln 1 −

𝐼0 𝜉ℎ + 𝑢 𝐿 = 0

𝐷0 =

ℎ−𝐼0/𝜉 ℎ+𝐼0

slide-63
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𝑳~𝑷(𝜻𝟑)

𝜻𝟑 𝜁 1 𝜁−1 𝐿

slide-64
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𝑳~𝑷(𝜻𝟑)

  • Again, vertical diffusion driven mixing occurs near instantaneously,

so the cleanser solution is well mixed vertically

𝜻𝟑 𝜁 1 𝜁−1 𝐿

slide-65
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𝑳~𝑷(𝜻𝟑)

  • Again, vertical diffusion driven mixing occurs near instantaneously,

so the cleanser solution is well mixed vertically

  • Horizontal diffusion allows reaction front to mix laterally

𝜻𝟑 𝜁 1 𝜁−1 𝐿

slide-66
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𝑳~𝑷(𝜻𝟑)

  • Again, vertical diffusion driven mixing occurs near instantaneously,

so the cleanser solution is well mixed vertically

  • Horizontal diffusion allows reaction front to mix laterally

𝜻𝟑 𝜁 1 𝜁−1 𝐿

  • Can reduce equations to:

𝜖 𝜖𝑢 ( ℎ + 𝐼0 𝐷0) = 𝜁2 𝜖 𝜖𝑌

ℎ + 𝐼0

𝜖𝐷0 𝜖𝑌

− 𝐿𝐷0,

𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝐷0

slide-67
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𝑳~𝑷(𝜻𝟑)

𝜻𝟑 𝜁 1 𝜁−1 𝐿

  • Can reduce equations to:

𝜖 𝜖𝑢 ( ℎ + 𝐼0 𝐷0) = 𝜁2 𝜖 𝜖𝑌

ℎ + 𝐼0

𝜖𝐷0 𝜖𝑌

− 𝐿𝐷0,

𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝑑0

slide-68
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𝑳~𝑷(𝜻𝟑)

𝜻𝟑 𝜁 1 𝜁−1 𝐿

𝐿 = 𝜁2, 𝜉 = 4

  • Can reduce equations to:

𝜖 𝜖𝑢 ( ℎ + 𝐼0 𝐷0) = 𝜁2 𝜖 𝜖𝑌

ℎ + 𝐼0

𝜖𝐷0 𝜖𝑌

− 𝐿𝐷0,

𝜖𝐼0 𝜖𝑢 = 𝐿𝜉𝑑0

slide-69
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Conclusions

slide-70
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Clean-ups with different reaction rates express different dynamics:

Conclusions

slide-71
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Clean-ups with different reaction rates express different dynamics:

  • Reaction occurs faster than vertical diffusion ⇒ Clean-up limited by

vertical diffusion rate.

Conclusions

slide-72
SLIDE 72

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Clean-ups with different reaction rates express different dynamics:

  • Reaction occurs faster than vertical diffusion ⇒ Clean-up limited by

vertical diffusion rate.

  • Reaction occurs slower than vertical diffusion ⇒ Cleanser is well

mixed vertically.

Conclusions

slide-73
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Clean-ups with different reaction rates express different dynamics:

  • Reaction occurs faster than vertical diffusion ⇒ Clean-up limited by

vertical diffusion rate.

  • Reaction occurs slower than vertical diffusion ⇒ Cleanser is well

mixed vertically.

  • Reaction occurs faster than horizontal diffusion ⇒ Cleanser

molecules only interact with agent beneath them.

Conclusions

slide-74
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Clean-ups with different reaction rates express different dynamics:

  • Reaction occurs faster than vertical diffusion ⇒ Clean-up limited by

vertical diffusion rate.

  • Reaction occurs slower than vertical diffusion ⇒ Cleanser is well

mixed vertically.

  • Reaction occurs faster than horizontal diffusion ⇒ Cleanser

molecules only interact with agent beneath them.

  • Reaction occurs as fast as horizontal diffusion ⇒ Cleanser

molecules can move horizontally to find agent.

Conclusions

slide-75
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Implications for Cleaning

slide-76
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  • For all reaction rates, should apply a thinner layer all over the agent

area for best reaction speed

Implications for Cleaning

slide-77
SLIDE 77

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  • For all reaction rates, should apply a thinner layer all over the agent

area for best reaction speed

  • For the slower reaction rate, 𝐿 ≈ 𝜁2, can decontaminate effectively

without knowing the exact agent shape

Implications for Cleaning

slide-78
SLIDE 78

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  • For all reaction rates, should apply a thinner layer all over the agent

area for best reaction speed

  • For the slower reaction rate, 𝐿 ≈ 𝜁2, can decontaminate effectively

without knowing the exact agent shape

Implications for Cleaning

slide-79
SLIDE 79

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  • For all reaction rates, should apply a thinner layer all over the agent

area for best reaction speed

  • For the slower reaction rate, 𝐿 ≈ 𝜁2, can decontaminate effectively

without knowing the exact agent shape

  • However, given enough time, any footprint of cleanser can

decontaminate completely given enough moles of cleanser

Implications for Cleaning