Numerical Treatment of
Thermophoretic Deposition
in Tube Flow
Dr. Patrick A. Tebbe (Faculty Advisor)
Minnesota State University, Mankato
Corey Thiebeault (Graduate Student)
University of Nevada, Reno
November 22, 2011
Presentation Intentions
1. Define thermophoretic deposition and
applications.
2. Review analytical and numerical approaches to
the problem.
3. Examine the problem’s complexity with various
numerical approaches.
Deposition Mechanisms
Deposition
Diffusion
Thermophoresis
Convection
Real World Applications
• Deposition in a tube approximates many modified CVD
and vapor axial deposition (VAD) processes; such as
production of fiber optic strands.
• Deposition of pollutants in the lung; such as Radon.
• Development of micro-electromechanical systems
(MEMS); for application and function.
• Soot deposition in exhaust systems; for purposes of
sampling and reduction.
• Nuclear power accidents; radioisotope transport in
existing and new reactor designs.
Real World Applications
Source: nasa.gov
Source: Gerd Keiser, Optical Fiber Communications,
2d ed., New York: McGraw-Hill, 1991.
Source: www.nrc.gov
Governing Equations

 u  0

 u   

2
 u  u    p    u  g
 o 
 t



 o  T  u   T
t

    k T



Navier-Stokes equations of continuity, conservation
of momentum, and energy.
Conservation of Species
1 
r u  V Tr n    v  V Tz n  1   rD n     D n 
r r
z
r r 
r  z 
z 
T
D
Cc
3 d p
  2.94 
V Tr
 
  lnT
 r
k k p   2.20Kn1  Kn1.2  0.41exp(0.88 / Kn)
1  3.438Kn1  2k k p   4.40Kn
Diffusive Deposition
P  1  5.5  2 / 3  3.77
for   0.007
P  0.819exp(11.5 )  0.0975exp(70.1 )
 0.0325exp(179 )

for   0.007
DL
Q
Isothermal diffusive transport to the walls (Hinds)
Graetz Problem

 ( x, r )  2 
exp(  2n x) F (r ,  n)
n 1
 n n
Laminar flow with a step change in temperature at the
wall. While a finite number of terms is needed they
can prove difficult to calculate. (Housiadas et al.)
The extended Graetz problem includes axial
conduction.
Boundary Layer Theory
Tube flow is split up into three distinct layers:
1. Convection dominated with no change in radial
concentration (core).
2. Thermophoretic layer with diffusion neglected.
3. Diffusion layer with convection neglected (wall).
Boundary Layer Theory
The Graetz solution is used to solve for
temperature (*). For PrK=1 the deposition
efficiency is found to be:
1  xˆ 


 ( xˆ)  (4.07)
*
1    Peth 
23
Where axial location and temperature are nondimensionalized (Williams and Loyalka)
Numerical and Experimental
Strattmann et al. studied cooled laminar tube flow
(30 to 100 nm particles)
• Found little influence by all axial effects (heat
conduction, diffusion, and thermophoresis)
• Material property changes were negligible
• An empirical formula was developed
NOTE: DT values were 20° to 100° C
Numerical and Experimental
Shimada et al. studied axially varying wall
temperatures (7 to 40 nm particles)
• Strong temperature influence on diffusion
coefficient was found.
• Empirical correlations show that
thermophoretic deposition cannot be
superposed on Brownian deposition.
NOTE: The maximum furnace temperature used
was 950° C
Numerical and Experimental
He and Ahmadi studied both laminar and
turbulent flows determining that:
• Smaller particles (0.01 m) are dominated by
diffusion.
• Larger particles (0.1 ≤ d ≤ 1 m) are dominated
by the thermophoretic force.
• Away from the wall, turbulence dominates
dispersion
• Near the wall, Brownian diffusion dominates
Numerical Approach #1
A de-coupled Eulerian approach was initially
chosen:
• Solution of flowfield and temperature in
FLOTRAN (ANSYS finite element module).
• Solution of species transport in separate finite
difference program (FORTRAN).
Numerical Approach #1
• Axi-symmetric geometry.
• Parabolic velocity profile at the inlet, zero
pressure at the outlet, no-slip on walls.
• Fluid assumed to be air with variable properties.
• Axial diffusion and thermophoresis are neglected.
• Negligible radial convection.
• Monodisperse inlet concentration of 1.0.
Program Verification
• The program was verified against the data of
Walker et al.
• Axial length set to 1 meter.
• Tube radius set to 0.01 meter.
• Particle diameters ranged from 1 to 50 nm.
Diffusion vs. Thermodiffusion
Inlet temperature = 293 K
Flowrate = 0.1 L/min
Particle diameter = 5 nm
Twall (K)
Pdiffusion
Pthermodiffusion
373
0.8425
0.8438
573
0.6825
0.6887
973
0.5566
0.5751
Note: P represents the “penetration” of particles.
Particle Diameter Effects
Penetration
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Thermophoretic
0.1
0
Diffusion only
0
10
20
30
40
50
Particle diameter (nm)
Inlet temperature = 293 K, Twall = 973 K
Flowrate = 0.5 L/min
Flowrate Effects
Percent change in penetration
(Inlet temperature = 293 K and particle diameter = 5 nm)
Twall (K)
0.1 L/min
0.5 L/min
373
0.15
0.85
573
0.9
2.22
973
3.32
6.98
(Pthermodiffusion – Pdiffusion)/Pdiffusion x 100%
Numerical Approach #2
A spectral collocation method has recently been explored:
• Spectral methods chose a basis function that is global
to the entire computational domain.
• Spectral methods select basis functions that are high
degree polynomials or trigonometric polynomials that
are infinitely differentiable.
• Progress to date has included studying the extended
Graetz problem and simple diffusion deposition.
Radial temperature profiles
Temperature profiles at different axial positions
Affect of axial conduction
Bulk fluid temperatures (non-dimensional) along
centerline of tube. Compared to a finite element
method on the right.
Affect of axial conduction (Pe=50)
Temperature profiles at different axial positions
Time comparison to finite difference
The finite difference method used a Jacobian
solution method.
The spectral method showed greater accuracy at
low grid size and shorter computation times.
Conclusions on spectral method
• The spectral collocation method showed good
agreement with other methods.
• The spectral method showed advantages in
terms of computational solution times.
• Its use would be limited by complex geometries.
Questions ?
References
He, C. and Ahmadi, G., 1998, “Particle Deposition with Thermophoresis in Laminar and Turbulent Duct
Flows,” Aerosol Science and Technology, 29, pp. 525-546.
Hinds, W.C., (1999). Aerosol Technology, 2nd Ed., John Wiley & Sons, New York.
Housiadas, C., Larrode, F. E., Drossinos, Y., (1999), Technical Note Numerical
Evaluation of the Graetz Series, Int. J. Heat Mass Transfer, Vol. 42, pp. 3013‐
3017.
H.-C. Ku and D. Hatziavramidis, “Chebyshev expansion methods for the solution of the extended
graetz problem," Journal of Computational Physics, vol. 56, no. 3, pp. 495 - 512, 1984.
M. . Y Bayazitoglu, “On the solution of graetz type problems with axial conduction," International
Journal of Heat and Mass Transfer, vol. 23, pp. 1399{1402, 1980.
S. Singh, “Heat transfer by laminar flow in a cylindrical tube," Applied Scientific Research, vol. 7, pp.
325-340, 1958.10.1007/BF03184993.
Shimada, M., Seto, T., and Okuyama, K., 1993, “Thermophoretic and Evaporational Losses of
Ultrafine Particles in Heated Flow,” AIChE Journal, 39, pp. 1859-1869.
Stratmann, F. and Fissan, H., 1988, “Convection, Diffusion and Thermophoresis in Cooled Laminar
Tube Flow,” Journal of Aerosol Science, 19, pp. 793-796.
Williams, M.M.R. and Loyalka, S.K., Aerosol Science: Theory and Practice, Pergamon, 1999.