Scraped Surface Heat Scraped Surface Heat ExchangersExchangers

B DuffyA D Fitt

M E-M LeeC P PleaseS K Wilson

Mathematics

U Blomstedt, N Hall-Taylor, J Mathisson

Industry

M J BainesD L PyleK-H Sun

Food BioscienceMathematics

H Tewkesbury

Technology Transfer

Overview of Current Overview of Current ResearchResearch

Problems in Fluid Dynamics Problems in Fluid Dynamics and Heat Transfer:and Heat Transfer:

Paradigm ProblemsParadigm Problems Channel flowChannel flow Thin cavityThin cavity

BladeBlade Affects of wear near the tip Affects of wear near the tip Stresses acting on BladeStresses acting on Blade

2D Flow2D Flow CavityCavity Inter-connected chambers Inter-connected chambers

3D Flow3D Flow

Mathematical Mathematical ConsiderationsConsiderations

Temperature dependant viscosityTemperature dependant viscosity Heat thinningHeat thinning

Non-Newtonian fluidNon-Newtonian fluid Power-law shear thinningPower-law shear thinning

Viscous DissipationViscous Dissipation

ConservationConservation MassMass MomentumMomentum EnergyEnergy

i

j

j

iij

q

ppq

m

nlnlT

kkkk

ik

m

nlnlT

kik

ik

i

k

k

x

u

x

ue

x

ueeee

x

Tk

xx

Tu

t

Tc

eeeexx

p

x

uu

t

u

x

u

2

1

2

2

0

2

1

2

1

Flow Around a BladeFlow Around a Blade

Problem FormulationProblem Formulation NewtonianNewtonian IsothermalIsothermal IncompressibleIncompressible Lubrication approximationLubrication approximation

y

v

x

u

y

p

x

p

y

u kkkkk

,0,2

2

L

h2

h1

Pivot

h0

x0

y

xU

y=0

y=H

blade

u1, p1, Q1

u2, p2, Q2

21

32

2

31

1

2

)(6

)3(6)(

QQUH

hH

hyyHQu

h

yhUhyQyhu

Pressure conditionsPressure conditions Far-field entry pressure must be equal to the pressures above and below Far-field entry pressure must be equal to the pressures above and below

the leading blade tipthe leading blade tip Far-field exit pressure must be equal to the pressures above and below the Far-field exit pressure must be equal to the pressures above and below the

trailing blade tiptrailing blade tip For the scraper to be in equilibrium, the moment about the pivot due to For the scraper to be in equilibrium, the moment about the pivot due to

pressure must vanishpressure must vanish

Blade AngleBlade Angle Independent of viscosity and the speed of the moving lower boundaryIndependent of viscosity and the speed of the moving lower boundary

0

00 ,h

H

L

xf

L

h

0 0x2

L L

No solutions when the No solutions when the

blade is pivoted near blade is pivoted near the trailing endthe trailing end

ExtensionsExtensions Shear-ThinningShear-Thinning Periodic blade-arraysPeriodic blade-arrays

21

21

)3(

6

h

yhyhUu

h

U

x

p

““Naïve” contact Naïve” contact problem has a singular problem has a singular forceforce

Asperities in blade and Asperities in blade and machine-casing machine-casing surfacessurfaces Solid-fluid contactSolid-fluid contact

Blade wear and Blade wear and geometrygeometry

Parallel Channel FlowParallel Channel Flow

0,0 uTT

Uuy

T

,0

21

2

2

1

0dy

du

dy

due

dy

Tdk

dy

du

dy

due

dy

d

dx

dp

m

T

m

T

hy

0y

energy

momentum

UnidirectionalUnidirectional SteadySteady Power-law fluidPower-law fluid Heat thinningHeat thinning Viscous Viscous

dissipationdissipation

Linear Stability AnalysisLinear Stability Analysis

21-m

21

2

2

221

1

1

1

1

1

1

arccosh

arccoshtanh2

arccoshtanh

arccoshtanh

1

arccoshcoshln

T

TmmT

T

T

T

e

ee

e

ey

u

eyTT

mk

hUT

TmTuUuyhy

mm

110

0,,

Thin Cavity ProblemThin Cavity Problem

Lubrication approachLubrication approach Large PecletLarge Peclet Small BrinkmanSmall Brinkman NewtonianNewtonian Neglect corner flowNeglect corner flow

0,

,,

,,

0

2

Teh

UlpT

TT

wl

Uhwv

l

UhvuUu

zh

Lzyhyxlx

222 BrPe

y

ue

y

T

yz

Tw

x

Tu

y

ue

yx

p

T

T

x

yz

0,,

,0,0

0

yUuTT

hyuy

T

Steady 2D ProblemSteady 2D Problem

FEMFEM ProblemsProblems

CavityCavity Annulus with bladesAnnulus with blades

ExtensionsExtensions 3D3D

i

j

j

iij

q

ppq

m

nlnlT

kkkk

ik

m

nlnlT

kik

ik

k

k

x

u

x

ue

x

ueeee

x

Tk

xx

Tu

eeeexx

p

x

uu

x

u

2

1

Pr

Br

Pr

1Re

Re

1

0

2

1

2

1

m=1.0 m=1.0 20% gap 60% gap

m=0.33 m=0.3320% gap 60% gap

Streamlines for 2D Cross SectionStreamlines for 2D Cross Section

1Pr 10Pr

210Pr 410Pr

m=1 Re=10 Br=0.3 b=0.05

Isotherms 2D Cross SectionIsotherms 2D Cross Section

Re=2, Br=0.512, Pe=3200, W/U=1/8, m=1.0

Isotherms for 3D Cavity ProblemIsotherms for 3D Cavity Problem

SummarySummary

Blade flowBlade flow Lubrication approximation and contact problemsLubrication approximation and contact problems Blade geometry and wearBlade geometry and wear

Paradigm problemsParadigm problems Stability problem for non-unique regimes in channel Stability problem for non-unique regimes in channel

flowsflows Slender cavity problems for a number of small Slender cavity problems for a number of small

parameter regimesparameter regimes Full two and three dimensional problemsFull two and three dimensional problems

Consolidate current findingsConsolidate current findings Numerical stability analysisNumerical stability analysis