Numerical Treatment of Thermophoretic Deposition in Tube Flow

1. 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

2. Presentation Intentions • Define thermophoretic deposition and applications. • Review analytical and numerical approaches to the problem. • Examine the problem’s complexity with various numerical approaches.

3. Deposition Mechanisms Deposition Diffusion Thermophoresis Convection

4. 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.

5. Real World Applications Source: nasa.gov Source: www.nrc.gov Source: Gerd Keiser, Optical Fiber Communications, 2d ed., New York: McGraw-Hill, 1991.

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

7. Conservation of Species

8. Diffusive Deposition Isothermal diffusive transport to the walls (Hinds)

9. 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.

10. Boundary Layer Theory • Tube flow is split up into three distinct layers: • Convection dominated with no change in radial concentration (core). • Thermophoretic layer with diffusion neglected. • Diffusion layer with convection neglected (wall).

11. Boundary Layer Theory The Graetz solution is used to solve for temperature (q*). For PrK=1 the deposition efficiency is found to be: Where axial location and temperature are non-dimensionalized (Williams and Loyalka)

12. 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

13. 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

14. Numerical and Experimental • He and Ahmadi studied both laminar and turbulent flows determining that: • Smaller particles (0.01 mm) are dominated by diffusion. • Larger particles (0.1 ≤ d ≤ 1 mm) are dominated by the thermophoretic force. • Away from the wall, turbulence dominates dispersion • Near the wall, Brownian diffusion dominates

15. 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).

16. 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.

17. 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.

18. Diffusion vs. Thermodiffusion Inlet temperature = 293 K Flowrate = 0.1 L/min Particle diameter = 5 nm Note: P represents the “penetration” of particles.

19. Particle Diameter Effects Inlet temperature = 293 K, Twall = 973 K Flowrate = 0.5 L/min

20. Flowrate Effects Percent change in penetration (Inlet temperature = 293 K and particle diameter = 5 nm) (Pthermodiffusion – Pdiffusion)/Pdiffusion x 100%

21. 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.

22. Radial temperature profiles Temperature profiles at different axial positions

23. Affect of axial conduction Bulk fluid temperatures (non-dimensional) along centerline of tube. Compared to a finite element method on the right.

24. Affect of axial conduction (Pe=50) Temperature profiles at different axial positions

25. 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.

26. 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 ?

27. 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.