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A Project Presentation for Applied Computational Fluid Dynamics By Reni Raju

A Project Presentation for Applied Computational Fluid Dynamics By Reni Raju

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## A Project Presentation for Applied Computational Fluid Dynamics By Reni Raju

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**Finite Element Model of Gas Flow**inside a Microchannel A Project Presentation for Applied Computational Fluid Dynamics By Reni Raju MECH - 523**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.**y**+H/2 x -H/2 Microchannel Experimental data from Pong et. al (1994) Numerical data from Chen et al (1998)**Governing Equations**Continuity Momentum Energy Equation of State**Normalized Equations**Continuity Momentum Energy Equation of State**Kn=0.0001**0.001 0.01 0.1 1 10 100 Continuum Regime Transition Regime Slip Flow Regime Molecular Regime Flow Regimes Knudsen Number Gad-el-hak (1999)**Wall Conditions**Wall Slip Maxwell (1879)**Thermal Boundary Condition**Temperature Jump Von Smoluchowski (1898)**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.**Numerical Details**Problem Statement Weak Statement Discrete Approximation Nk is appropriate basis function; Chebyshev , Lagrange or Hermite interpolation polynomials complete to degree k.**FE Formulation**Momentum Equation Variable Discretized Weak Statement**1**7 4 3 6 8 9 1 2 5 Discretization FE Basis Cartesian Coordinate**Global/Local frame**Element Jacobian Differential Element Diffusion Term Matrix Form**Solution Procedure**FE Formulation Time Integration Form NR Iteration Convergence Criteria = 10-4 for all integrated quantities.**1**7 4 3 6 8 9 1 2 5 Mesh • FE basis • 2D-Quadratic 9-node • 2D-Bilinear 4-node • Mesh • 1369 nodes • 324 elements 2 1 4 3 2 1 2**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 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 enddo**Boundary conditions**• At the Inlet • The Gas temperature Ti is specified as 314 K . • The y-component of the velocity v = 0 . • Inlet pressure, Pi is specified based on the corresponding Pressure ratio. • At the Outlet • The pressure at the outlet, P0 is 100.8 KPa. • On the Walls (No-slip) • For isothermal wall the wall temperature Tw is 314 K. • u and v velocity components = 0.**Microchannel Flow Contours**Channel Aspect Ratio: 2500 Velocity Pressure Pressure ratios: 1.340, 1.680, 2.020, 2.361, 2.701**Slip/No-Slip Comparison**Pressure Velocity Slip variation up to ~ +4% Slip variation up to ~ +8%**Pressure Distribution(Numerical-Experimental Comparison)**Experimental validation within ~ 4 % Numerical validation within ~ 1.3 %**Velocity Profile(Numerical-Numerical Comparison)**Numerical validation within ~ 2.5 %**y**+H/2 x -H/2 Conclusion • For Microchannel- • Finite Element Model compares well with reported Numerical and Experimental results. • The slip conditions show higher flow rates.**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).**Acknowledgements**• Dr. Subrata Roy. • Dr. Birendra Pandey. • Center of Nanotechnology, NASA Ames. • NSF/NPACI Supercomputer. • Electric Propulsion Laboratory, NASA Glenn Research Center.