1. 1 A Project Presentation for Applied Computational Fluid Dynamics By Reni Raju Finite Element Model of Gas Flow inside a Microchannel MECH - 523

2. 2 Objectives Develop a 2D Finite Element Model for gas flow inside a microchannel.  Develop FE formulation for N-S equations.  Implement Slip and temperature boundary conditions.  Compare Numerical and Experimental Data for Microchannel.

3. 3 Microchannel ParameterRange or Mean Value Length L 3000  m Width W 40  m Height H 1.2  m Pressure Ratio1.340, 1.680, 2.020, 2.361, 2.701 Inlet Temperature T 0 314 K Wall Temperature T w 314 K Knudsen Number, (Kn= l /L)0.055 Abs. Viscosity  1.85 x 10 -5 Ns/m 2 Spec. Gas Constant (N 2 ) R296.7 J/kg K Ratio of Spec. heats  1.4 Numerical data from Chen et al (1998) +H/2 - H/2 y x Experimental data from Pong et. al (1994)

4. 4 Governing Equations Continuity Momentum Energy Equation of State

5. 5 Normalized Equations Continuity Momentum Energy Equation of State

6. 6 Flow Regimes Kn=0.00010.0010.010.1110100 Continuum Regime Slip Flow Regime Transition Regime Molecular Regime Knudsen Number Gad-el-hak (1999)

7. 7 Wall Conditions Wall Slip Maxwell (1879)

8. 8 Thermal Boundary Condition Temperature Jump Von Smoluchowski (1898)

9. 9 Finite Element Algorithm Developed by the Computational Plasma Dynamics Laboratory at Kettering University (Roy, CMAME, v184, 87-98, 2000). A Family of complex subroutines that can study macroscopic collisional plasmas. (Roy and Pandey, POP, v9, 4052-60, 2002). Written in Fortran 77, use Cray-style Fortran pointers, and are designed for UNIX-type environment. Two dimensional formulation (so far). Implemented Sub-Grid Embedded (SGM) FE for Coarse-grid solution Stability,Accuracy and Tri-diagonal Efficiency (Roy and Baker, NHT-B, v33, 5-36, 1998). Utilized to model Compressible flow through Electric Propulsion thrusters including Microchannels.

10. 10 Numerical Details Weak Statement Discrete Approximation N k is appropriate basis function; Chebyshev, Lagrange or Hermite interpolation polynomials complete to degree k. Problem Statement

11. 11 FE Formulation Momentum Equation Variable Discretized Weak Statement

12. 12 Discretization FE BasisCartesian Coordinate 1 4 2 3 7 8 6 5 11 9

13. 13 Global/Local frame Diffusion Term Matrix Form Element Jacobian Differential Element

14. 14 Solution Procedure NR Iteration Convergence Criteria = 10 -4 for all integrated quantities. FE Formulation Time Integration Form

15. 15 Mesh FE basis  2D-Quadratic 9-node  2D-Bilinear 4-node Mesh  1369 nodes  324 elements 1 4 2 3 7 8 6 5 11 9 1 4 2 3 11 22 22

16. 16 Code Format do i = 1,4 do j = 1,9 ii = loc_rho(i) jj = loc_v1(j) term = wt * rho * Nmat22(i) * DNmat33Dx(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term jj = loc_v2(j) term = wt * epsi * rho * Nmat22(i) * DNmat33Dy(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term enddo do i = 1,4 do j = 1,4 ii = loc_rho(i) jj = loc_rho(j) term = wt * velx * Nmat22(i) * DNmat22Dx(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term jj = loc_rho(j) term = wt * epsi * vely * Nmat22(i) * DNmat22Dy(j) elemk_lin(ii,jj) = elemk_lin(ii,jj) + term enddo

17. 17 Boundary conditions At the Inlet  The Gas temperature T i is specified as 314 K.  The y-component of the velocity v = 0.  Inlet pressure, P i is specified based on the corresponding Pressure ratio. At the Outlet  The pressure at the outlet, P 0 is 100.8 KPa. On the Walls (No-slip)  For isothermal wall the wall temperature T w is 314 K.  u and v velocity components = 0.

18. 18 Pressure Microchannel Flow Contours Velocity Pressure ratios: 1.340, 1.680, 2.020, 2.361, 2.701 Channel Aspect Ratio: 2500

19. 19 Slip/No-Slip Comparison Slip variation up to ~ +8% Velocity Slip variation up to ~ +4% Pressure

20. 20 Pressure Distribution (Numerical-Experimental Comparison) Experimental validation within ~ 4 % Numerical validation within ~ 1.3 %

21. 21 Velocity Profile (Numerical-Numerical Comparison) Numerical validation within ~ 2.5 %

22. 22 Conclusion For Microchannel-  Finite Element Model compares well with reported Numerical and Experimental results.  The slip conditions show higher flow rates. +H/2 - H/2 y x

23. 23 Future Scope Extend to the applicability of Slip-flow conditions to the Transition regime, ( 0.1 < Kn < 10 ). Applicability to higher Knudsen number ranges (Nanoscale devices).

24. 24 Acknowledgements Dr. Subrata Roy. Dr. Birendra Pandey. Center of Nanotechnology, NASA Ames. NSF/NPACI Supercomputer. Electric Propulsion Laboratory, NASA Glenn Research Center.