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