1. Scraped Surface Heat Exchangers B Duffy A D Fitt M E-M Lee C P Please S K Wilson Mathematics U Blomstedt, N Hall-Taylor, J Mathisson Industry M J Baines D L Pyle K-H Sun Food Bioscience Mathematics H Tewkesbury Technology Transfer

2. Overview of Current Research Problems in Fluid Dynamics and Heat Transfer: Paradigm Problems Paradigm Problems Channel flow Channel flow Thin cavity Thin cavity Blade Blade Affects of wear near the tip Affects of wear near the tip Stresses acting on Blade Stresses acting on Blade 2D Flow 2D Flow Cavity Cavity Inter-connected chambers Inter-connected chambers 3D Flow 3D Flow

3. Mathematical Considerations Temperature dependant viscosity Temperature dependant viscosity Heat thinning Heat thinning Non-Newtonian fluid Non-Newtonian fluid Power-law shear thinning Power-law shear thinning Viscous Dissipation Viscous Dissipation Conservation Conservation Mass Mass Momentum Momentum Energy Energy

4. Flow Around a Blade Problem Formulation Problem Formulation Newtonian Newtonian Isothermal Isothermal Incompressible Incompressible Lubrication approximation Lubrication approximation L h2h2 h1h1  Pivot h0h0 x0x0 y x U y=0 y=H blade u 1, p 1, Q 1 u 2, p 2, Q 2

5. Pressure conditions Pressure conditions Far-field entry pressure must be equal to the pressures above and below the leading blade tip Far-field entry pressure must be equal to the pressures above and below the leading blade tip Far-field exit pressure must be equal to the pressures above and below the trailing blade tip Far-field exit pressure must be equal to the pressures above and below the trailing blade tip For the scraper to be in equilibrium, the moment about the pivot due to pressure must vanish For the scraper to be in equilibrium, the moment about the pivot due to pressure must vanish Blade Angle Blade Angle Independent of viscosity and the speed of the moving lower boundary Independent of viscosity and the speed of the moving lower boundary

6. No solutions when the blade is pivoted near the trailing end No solutions when the blade is pivoted near the trailing end Extensions Extensions Shear-Thinning Shear-Thinning Periodic blade-arrays Periodic blade-arrays “Naïve” contact problem has a singular force “Naïve” contact problem has a singular force Asperities in blade and machine-casing surfaces Asperities in blade and machine-casing surfaces Solid-fluid contact Solid-fluid contact Blade wear and geometry Blade wear and geometry

7. Parallel Channel Flow energy momentum Unidirectional Unidirectional Steady Steady Power-law fluid Power-law fluid Heat thinning Heat thinning Viscous dissipation Viscous dissipation

8. Linear Stability Analysis

9. Thin Cavity Problem Lubrication approach Lubrication approach Large Peclet Large Peclet Small Brinkman Small Brinkman Newtonian Newtonian Neglect corner flow Neglect corner flow x y z

10. Steady 2D Problem FEM FEM Problems Problems Cavity Cavity Annulus with blades Annulus with blades Extensions Extensions 3D 3D

11. m=1.0 20% gap 60% gapm=0.33 Streamlines for 2D Cross Section

12. m=1 Re=10 Br=0.3 b=0.05 Isotherms 2D Cross Section

13. Re=2, Br=0.512, Pe=3200, W/U=1/8, m=1.0 Isotherms for 3D Cavity Problem

14. Summary Blade flow Blade flow Lubrication approximation and contact problems Lubrication approximation and contact problems Blade geometry and wear Blade geometry and wear Paradigm problems Paradigm problems Stability problem for non-unique regimes in channel flows Stability problem for non-unique regimes in channel flows Slender cavity problems for a number of small parameter regimes Slender cavity problems for a number of small parameter regimes Full two and three dimensional problems Full two and three dimensional problems Consolidate current findings Consolidate current findings Numerical stability analysis Numerical stability analysis