Numerical Modeling of Flow and Transport Processes Simulating Laminar Natural Convection on vertical plate using Comsol

1. Numerical Modeling of Flow and Transport Processes Simulating Laminar Natural Convection on vertical plate using Comsol Software Joseph Addy 31103196 Supervised by Professor Manfred Koch

2. Motivation • Introduction to COMSOL Software • Natural Convection and Governing Equations • Boundary and Initial conditions for Problem description • Comsol Implementation • Conclusion Table of Contents

3. Simulation is fast and delivers local and global information in details, for example the velocity field or pressure distribution in a pipe. • Simulation provides approximate solution to real problems and also cheaper. • Numerical modeling provides general knowledge and understanding for the mode of operation of products and processes Motivation

4. COMSOL Multiphysics software is a powerful finite element (FEM), partial differential equation (PDE) solution engine. The software has eight add-on modules that expand the capabilities in the following application areas: • AC/DC Module • Acoustics Module • Chemical Engineering • Earth Science • Heat transfer • MEMS (microelectromechanical systems) • RF • Structural Mechanics • The Multiphysics has CAD Import Module and the Material Library as supporting software COMSOL Software

5. COMSOL Software Multiphysics implies multiple branches of physics can be applied within each model. Comsol facilitates all steps in the modeling begining with your geometry, meshing , specifying your physics ( subdomain and boundary conditions), solving and then visualizing your results. Comsol can solve problems in 1D, 2D, 3D and axial symmetries in 1 and 2D

6. Comsol Multiphysics provides the power to simulate and model systems that involve • Mass transport including migration in the concentrated solutions • Laminar, turbulent , multiphase and porous media flows • Energy transport and thermodynamic systems Comsol Multiphysics

7. From the application modes one chooses the type of problem to be solved and from the space dimension, the type of dimension. Comsol software

8. Natural Convection refers to convection in which the fluid is driven thermally; that is , fluid motion is driven by density gradients that is induced in the fluid whether it is heated or cooled. The velocities induced by the density gradients are small and therefore the natural heat transfer coefficient are lower than forced convection. The governing differential equations for natural convection are based on the conservation of mass, momentum and thermal energy. The density difference in natural convection is driven by a temperature difference and known as Boussinesq aprroximation( where the density is assumed to only depend on temperature). Natural Convection (free convection)

9. Governing Equations The left part of the momentum and energy equations consist of the convective and local part of the derivative

10. For incompressible Newtonian fluid: the density was held constant and the shear stress is proportional to the velocity gradient. Governing Equations Principle of mass conservation Principle of conservation of momentum for incompressible fluid. The second law of Newton states that mass multiplied by acceleration is equal to the sum of forces acting on the body

11. Governing Equations Dissipation , source and sink are neglected Energy cannot be created nor destroyed, but can be transformed from one state to another ρ: density cp: specific heat capacity η: dynamic viscosity P: pressure f: volume force sp(T.D): dissipation energy T: temperature u: velocity in x direction v: velocity in y direction div: vector differential operator

12. u=v=0 T=0°C Problem description with boundary and initial conditions The momentum and energy equations are inherently coupled since the temperature appears in the momentum equation and the velocity in the energy equation. u=v=0 T=20°C u=v=0 T=70°C u=v=0 T=0

13. β: the thermal expansion coefficient υ: kinematic viscosity The solution for natural convection is correlated using the Rayleigh number and Nusselt number. The Rayleigh number determines whether the free convection is laminar or turbulent at the critical boundary layer.

14. The Navier-Stokes equation is coupled with the heat equation From the Applications Mode , choose Comsol Multiphysics Add heat transfer to Fluid dynamics Comsol Implementation

15. In the option builder the physical quantities used in the computation are expressed on the average fluid temperature Comsol Implementation

16. Comsol Implementation The physical quantities and the boundary conditions are fixed

17. Comsol Implementation Mesh containing 590 elements Mesh containing 2360 elements

18. Temperature Surface

19. Velocity field within a surface temperature

20. Streamline of velocity field

21. For steady state problems coupling the fluid dynamics and heat transfer with no slip boundary conditions , the fluid moves in a direction of low temperature as been expected and also from kinetic theory the temperature is proportional to the velocity. The streamline diagram of the velocity field shows vorticity in the direction of decreasing temperature. • The Nusselt number is a function of the Rayleigh`s number, therfore the higher the natural heat transfer coefficient. Conclusion

22. Heat transfer in condensation and Boiling : Karl Stephan • Convective Boiling and Condensation: Collier and Thome • Heat transfer: Gregory Nellis and Sanford Klein • Matlab for poisson equation: Professor Koch • Multiphysics modeling Comsol: Roger W. Pryor • Stanford University Lecture: Prof.Barba Lorenza • MIT matlab code for incompressible navier stokes equation: Prof. Gilbert Strang References

23. Thank you for your attention