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

Diffusion

Deposition

ConvectionThermophoresis

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

Navier-Stokes equations of continuity, conservation of momentum, and energy.

0 u

gupuut

uo

2

TkTut

To

Conservation of Species

z

nD

zr

nDr

rrnVv

znVur

rrTzTr 11

r

TV Tr

ln

C

d

TD c

p

3

KnkkKn

KnKnKnkk

p

p

40.421438.31

)/88.0exp(41.02.1120.2

94.2

Diffusive Deposition

0.007for 77.35.51 3/2 P

0.007for )179exp(0325.0

)1.70exp(0975.0)5.11exp(819.0

P

Q

DL

Isothermal diffusive transport to the walls (Hinds)

Graetz Problem

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.

1

2 ),()exp(2),(n nn

nn rFxrx

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:

Peth

xx

ˆ

1

1)07.4()ˆ(

32

*

Where axial location and temperature are non-dimensionalized (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: T 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

Inlet temperature = 293 K, Twall = 973 K

Flowrate = 0.5 L/min

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Particle diameter (nm)

Pe

ne

tra

tio

n

Thermophoretic

Diffusion only

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 NumericalEvaluation 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.