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