Mathematical Modelling in Industry

Nev Fowkesneville.fowkes@uwa.eu.au

School of Maths an StatsUniversity of Western Australia

November 10, 2012

Industrial Mathematics: MISGs

I am a industrial mathematics modeller.MISG (Maths in Industry Study Groups)(www.maths-in-industry.org):Originated at Oxford 35 years ago. Now worldwide (about 16 eachyear) and linked. I have attended/help facilitate more than 50 ofthese in Australia, Asia, Africa, Canada, Europe, UK.The format: problems presented by industry on Monday, resultspresented by academics on Friday. Followup: proceedings producedlater.These are worldwide interconnected activities (about 16 each year).The next MISG in Australia is at QUT in Feb 2013.

Problem Areas

I Environmental Areas: fire (home, forests, caused by electricwires), hydraulics, waste disposal

I Energy: wind turbines, alcohol from cellulose, coal extraction,explosives,

I Operations research; optimal ore extraction, scheduling(airlines), water management.

I Medical: Cancer research, epilepsy, lens manufacture (coatingsfor lenses, manufacture of plastic lenses), scheduling patients.

I Mining/Construction: ore detection, extraction. explosives,safety (tunnel collapse), corrosion, concrete.

I Sports and Defence: soccer, badminton racquets, computerand defence games.

I Agriculture: tea drying, sprays for crops, labelling of bottles,sugar beer production.

I Manufacturing: continuous casting of steel, corrosion in sheetsteel, wine bottle labels, icecream manufacture, beerproduction, paper (rolls, sheets).

I Finance (bidding for energy supply), product differentiation.

I Transport: air rail scheduling, noise reduction, mechanicalproblems.

So what does a industrial mathematical modeller have tooffer?

As a mathematical modeller I can say that I am expert in none ofthe disciplines required to understand such problems.So what skills are offered by a modeller?

I Cross disciplinary modelling experience: the ‘same’ problemarises in many contexts!

I Mathematical technique (scaling, computational, asymptotics)

The primary issues are not numeric: the aim is to understand thescience.Mathematics is just one tool for doing thisNow for a few problems:

An Industrial Packer (Inflatable Packers International,Perth, Enterprise Connect)

Inflatable packers are used for sealing off sections of a drill hole inthe oil and gas and geothermal industries (permeability, sampling).

borehole

The reinforced rubber tube

A packer needs to be easily inflated but able to withstand highpressure once in place.This is achieved using wire cords are imbedded in rubber.The cord winding angle changes during inflation.

600mm

60mm

Issues: practical and technical

Commercial Issues:

I Hand crafted.

I Often need to be designed (effective sealing and rupture) for aparticular application.

I Need to be reliably designed (strict design specifications).

Practical/Technical Issues:

I Winding angle issues

I Rubber issues

I Scaling issues

I Composite material issues

All these issues are subtle (crude computation is useless).Understanding is essential.

Continuous Casting of Steel (BHP/Australia, Voest)Normally steel is cast into blocks (20cms thick) and then rolled; avery expensive process.In continuous steel casting a strand of molten steel movesdownwards under gravity and is cooled by water filled moulds.

I The moulds are oscillated. (what amplitude, frequency?)I Flux facilitates the process (insulation, lubrication properties?)

molten steel

V

a sin wt

solidsteel

water

flux

5m

cooled

0.2m

Figure : Strand Casting: The strand moves downwards under gravity.

Problems

The oscillating mould feeds flux into the gap.Major problems arise if flux doesn’t fill the gap (thickness 1-2mm).

water

flux

molten steel U

solid

V

downstroke: flux movement blocked

upstroke: flux drawn into the gap

water

flux

molten steel

U

solid

V

The essence: the molten steel acts as a pump; sucking flux intothe gap (upstroke) and then blocking the flow (downstroke).Subtle!

Steel sheet casting BHP

In this process a thin sheet (1cm) of molten steel is poured onto arotating cylinder and a steel sheet is drawn off.No rolling needed!What radius is required to ensure solidification without spilling?

2mm

3mwater

molten steel

solidified steel

The essence lies in the determination of the long molten tail.

Capacitor Dipping: DuPont

Silica ‘slabs’ (1mm by 2mm) are dip coated in metal.Question: How to reduce mooning (surface tension)?

The essence lies in understanding corner effects; really importantfor all industrial coating problems (Blue Steel, lens coatings).Major understanding issues remain. Computationally difficult.

Rock Blast prevention: Thin Spray Liners. (S Africamining)

It has been observed that thin (1mm) elastically weak spray linershelp stabilise tunnel walls (prevent rock blasts).Why?

Tunnel collapse: Crack Extension

The essence: The stress singularity at the the crack tip is reducedby the presence of the liners:

τij =1

r1/2(cos(θ/2), sin(θ/2)) → τij =

1

rγ(cos(γθ), sin(γθ))

where r is the distance from the crack tip. γ → 0 for even weakliners.

I Thus cracks don’t extend.I Key stones don’t fall out.I Tunnels don’t collapse under the action of (seismic and

steady) loading.

Primary requirement of the liners are small viscosity, good adhesivestrength (elastic strength unimportant).

Earthquakes and Mining (S Africa)Can water draining into the deep (2000m) disused gold mines leadto increased earthquakes?

The essence:

I It takes about 1 year for water to enter available faults.

I The effect is to increase the prevalence of earthquakes by10-20% due to hydrostatic loading and (unexpectedly) the slipangle range is increased.

Johannesburg look out

Concrete Problems

I Dam construction: cooling using water pipes (network design)

I Maturity (chemistry, cracking)

I Concrete cancer (chemistry, cracking)

Icecream (Unilever UK)Icecream is a really quite remarkable composite material: There aresolid ice crystals, water in a viscous sugary liquid, and fat uniformlyspread, and coexisting (liquid, solid, gaseous) in a delicious smoothand stable product.One can apply material science techniques to produce the icecreams of the future (sintering, particle growth).

Figure : Figure Title

The problem is with stability/commerce

The essence: To make more money one needs to increase theamount of air and water and decrease the fat content whichcompromises stability/taste. Additives (surfactants) are used tostabilise the ‘foam’.Problems of interest:

I Quality control (additives) under commercial/transportconditions (excess crystal growth, bubbles bursting).

I Designer ice creams. (would like to produce special shapes toattract customers)

I Chocolate coatings

Wine Labels (Australia, S Africa): a lubrication problemHow to prevent bubbles in the label?

Figure : Wrinkling of wine labels (Southcorp)

Soccer ball design (SAfrican Soccer Assn)

World Soccer Championships 2010 (Johannesburg (2000m))Can one choose a soccer ball design to advantage S Africa?

Answer: Yes and they did.It’s all about swerve: reverse and direct Magnus effect.

I Johannesburg is 2000m above sea level (air density is 20%below that of sea level.)

I Forces acting on a movie ball are gravity and air drag.

I Spin produces greater drag on one side of the ball producingswerve (Magnus effect)

I At low speeds there is a transition in the air flow at around10-15 m/sec which leads to a reverse Magnus effect.

I The effect is much greater if the ball is smoother and lighterand stitching is minimal especially under low densityconditions. Perturbation procedures produce explicit results.

I Yes nonlocal players had trouble playing in Johannesburg.