numerical modelling of scraped surface heat exchangers k.-h. sun 1, d.l. pyle 1 a.d. fitt 2, c.p....

Numerical Modelling of Scraped Surface Heat Exchangers

K.-H. Sun1, D.L. Pyle1

A.D. Fitt2, C.P. Please2

M. J. Baines3, N. Hall-Taylor4

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

Using asymptotic and numerical methods, to explore selected sub-problems relating to SSHE design and performance.

Asymptotics: simplified problems understanding

Numerics: complex problems, effects of geometry etc

Schematic & Coordinate System(fixed to inner surface)

Singularity here

Objectives – this study

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

EquationsThe non-dimensional form of the steady two-

dimensionalequations of an incompressible fluid are

The frame of reference is rotating in the z direction: the Coriolis force term is added

2

0

2

0

)(1

)(Re

1

Re

12

0

IPe

BrT

PeTv

vpvkvv

v

Modifications to power law Viscosity (modified power law):

m – typically ca. 0.33 – shear thinning b – heat thinning index

c ensures that the viscosity is non-zero when I2 approaches infinity at the singularity.

In the stagnation areas, I2 is monitored against a minimum value of 0.000001 to ensure that the viscosity is finite.

2222

2/)1(20

)()(2)(2

)(

x

v

y

u

y

v

x

uI

cIe mbT

Dimensionless Groups

nd

Td

Tk

qL

k

hLTk

Uor

k

Ub

kULck

c

UL

p

p

Nu

)(Br

/Pr.RePe

Pr

Re

20

200

0

0

Reynolds number

Prandtl number

Peclet number

Brinkman number

Nusselt number

FEM solution procedureAll 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.

nnn

nnnnnn

vPenpp

vkvvvpvPen

1

211 02

Re

1)(

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 10128 nodes and 4724 6-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% gap 20% gap 60% gap

m=0.33 (Shear Thinning) 0% gap 20% gap 60% gap

Streamfunctions: isothermal flow

Results:

Increasing gap removes stagnation zone upstream of blade

Increasing gap width shifts stagnation point downstream of blade

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

Checked convergence etc with mesh size and configuration

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=0.330% gap 0% gap

20% gap 20% gap

Temperature profiles near the blade [Values = T-Twall]

Max Temp

MaxTemp

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 heat thinning on heat transfer

1

10

100

1000

0 0.2 0.4 0.6 0.8 1Normalized distance x

Loca

l hea

t flu

x m=1.0

m=1.0T

m=0.33

m=0.33T

Effect of shear thinning and heat thinning on integrated heat transfer:

Increasing “b” corresponds to increased heat thinningRe=10 Pr=10 Br=1.0, 20% gap (Straight blade)

Effect of heat thinning on heat transfer

0

5

10

15

20

25

0.2 0.4 0.6 0.8 1Power law index m

Wal

l hea

t flu

x

b=0.0

b=0.05

b=0.1

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 gap size on heat transfer

0

2

4

6

8

0.2 0.4 0.6 0.8 1

Power law index m

Wal

l hea

t flu

x

0%g

20%g

60%g

100%g

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]

Effect of m on heat transfer

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

Normalized distance x

Loca

l hea

t flu

x

m=0.33

m=0.6

m=1

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

Nusselt number for tangential flow in an annulus

00.5

11.5

22.5

3

0.2 0.4 0.6 0.8 1

Power law index

Nu

Nuc1

Nu1

Nuc0.3

Nu0.3

Velocity profile for tangential flow in an annulus

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Normalized distance

velo

city

m=0.33

m=0.33c

m=0.6

m=0.6c

m=1.0

m=1.0c

Convergence Convergence

The procedure was coded in fastflo (a commercial FEM PDE solver)

nnn

nnn

TTT

UUU

1

1

For large power law index m>=0.4

For small power law index m<0.4

000001.0,000001.0

0001.0,0001.0

Mesh- thermal calculations

The mesh has 14912 nodes and 6908 6-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