Numerical Modelling of Scraped Surface Heat Exchangers K.-H. Sun 1, D.L. Pyle 1 A.D. Fitt 2, C.P. Please 2 M. J. Baines 3, N. Hall-Taylor 4 1 School of Food Biosciences, University of Reading, RG6 6AP, UK 2 Faculty of Mathematical Studies, University of Southampton, SO17 1BJ, UK 3 Department of Mathematics, University of Reading, RG6 6AP, UK 4 Chemtech International Ltd, Reading, RG2 0LP, UK

Outline of Presentation  Background and objectives  Equations and boundary conditions  Isothermal results  Heat transfer results  Conclusions

Background  Scraped surface heat exchangers are widely used in the food industry for processing highly viscous, shear-thinning fluids (margarine, gelatine etc..).

Background  The cold outer cylinder is scraped periodically by the blades to prevent crust formation and promote heat transfer.  With a rotating inner cylinder, the flow is a superposition of Poiseuille flow in an annular space and a Couette flow.  Typical flows are non-Newtonian with low Reynolds and high Prandtl numbers.  Poor understanding of mechanisms: no rigorous methods for design, optimisation and operation.

Objectives - overall w Using asymptotic and numerical methods, to explore selected sub-problems relating to SSHE design and performance. w Asymptotics : simplified problems understanding  Numerics: complex problems, effects of geometry etc

Schematic & Coordinate System (fixed to inner surface) Singularity here

Objectives – this study w FEM numerical modelling studies of two-dimensional steady problems : 1. Isothermal behaviour – effects of blade design 2. Local and integrated heat transfer & effect of: Blade design Power law index – i.e Shear thinning Heat thinning

Equations The non-dimensional form of the steady two- dimensional equations of an incompressible fluid are The frame of reference is rotating in the z direction: the Coriolis force term is added

Modifications to power law  Viscosity (modified power law):  m – typically ca – shear thinning  b – heat thinning index c ensures that the viscosity is non-zero when I 2 approaches infinity at the singularity. In the stagnation areas, I 2 is monitored against a minimum value of to ensure that the viscosity is finite.

Dimensionless Groups Reynolds number Prandtl number Peclet number Brinkman number Nusselt number

FEM solution procedure All problems were solved using FASTFLO, a commercial FEM solver. The isothermal flow was solved using the FEM augmented Lagrangian method; the iterative procedure for Newtonian fluid was For the non-isothermal condition: 1. The velocity was solved by assuming a fixed temperature field 2. Then the temperature field was solved from the known velocity 3. The procedure was repeated until a converged velocity and temperature was reached.

Problem 1 :Isothermal case  Boundary conditions: zero slip all surfaces  Streamlines, stagnation line, pressure distribution etc  Effect of shear thinning (“m”)  Effect of blade design: flow gap, angle:

Mesh – Isothermal problem (Straight blades)  The mesh has nodes and node triangle elements  It is concentrated along the blades and the tip of the blades.  Singularity at tip: add 2% “tip gap” with at least 5 mesh points

Streamlines-- Effect of gap size with angled blade (AB) Re = 10 m=1.0 (Newtonian) 0% gap20% gap60% gap m=0.33 (Shear Thinning) 0% gap20% gap60% gap

Streamfunctions: isothermal flow

Results: w Increasing gap removes stagnation zone upstream of blade w Increasing gap width shifts stagnation point downstream of blade w Increased shear thinning (i.e. lower m) shifts stagnation line

Problem 2: Heat transfer  Boundary conditions  Credibility check  Temperature contours (straight blade)  Heat flux at the wall: local and integrated Effect of shear thinning Effect of heat thinning Effect of blade design: gap, angle

Thermal boundary conditions

Credibility checks w Checked convergence etc with mesh size and configuration w Checked against analytical results for simplified problems (eg flow in annulus)

Temperature contours Re=10 Pr=10 Br=0.3, b=0.05 NEWTONIAN SHEAR THINNING m=1 m=0.33 0% gap 20% gap

Temperature profiles near the blade [Values = T-T wall ] Max Temp Max Temp

Effect of heat thinning on heat transfer Re=10 Pr=10 Br=1.0, b=0.0 or 0.1, 20% gap (T corresponds to b = 0.1 – ie heat thinning)

Effect of shear thinning and heat thinning on integrated heat transfer: Increasing “b” corresponds to increased heat thinning Re=10 Pr=10 Br=1.0, 20% gap (Straight blade)

Conclusions  For a constant viscosity fluid the highest shear region is close to the blade tip; this gives rise to high viscous heating; the maximum temperature and heat flux areclose to the tip  For shear thinning fluids, the viscosity is reduced in the high shear region, so viscous heating is reduced together with the heat flux (Nusselt number)  The heat thinning effect is more significant for Newtonian fluids; it also further reduces viscous heating, local hot spots and the heat flux. .

Future work  Address 3-D problem – in first instance by “marching 2-D” approach  Address numerical problems at high Pr  In parallel: analytical approach to selected sub- problems  Produce solutions to engineering problems eg:  Blade force and wear  Power requirements & Heat transfer  Mixing

Acknowledgements  The authors wish to acknowledge support from The University of Reading and Chemtech International.

Additional information

Conclusions – 1 - methodology  The FEM method gives good agreement with analytical results where comparison is possible  Results are robust to changes in mesh size etc  A small (fictitious) gap between the tip and wall helps avoid numerical problems due to the singularity at the tip/wall intersection  Need much finer mesh grid at very high Prandtl numbers

Conclusions - 2  The flow gap acts to:  Release the stagnation area near the foot of the blades,  Reduce the force on the blades and  Shift the location of the centre stagnation point.

Effect of gap size on heat transfer (straight blade) Re=10 Pr=10 Br=0.3, b=0.05

Effect of power law index m on local heat transfer across cold surface Re=10 Pr=10 Br=0.3, b=0.05, 20% gap (straight blade) [m = 1: Newtonian; m < 1: Shear thinning]

Tangential flow in an annulus: comparison with analytical solution Re=10 Pr=10 Br=0.3, b=0.0 (no heat thinning) c-analytical value Velocity profile Nusselt number

Convergence  Convergence  The procedure was coded in fastflo (a commercial FEM PDE solver) For large power law index m>=0.4 For small power law index m<0.4

Mesh- thermal calculations  The mesh has nodes and node triangles  It is concentrated along the surfaces and on a line along the blades.  There is a 2% gap at the tip with at least 5 mesh points