README.md

ATP: Advanced Transport Phenomena

Collection of numerical codes presented in the Advanced Transport Phenomena (ATP) class at Politecnico di Milano.

Description

The numerical codes are organized in the following folders:

1. lectures: codes presented and discussed in the ATP lectures

2. practicals: codes and problems presented and discussed in the ATP practical sessions (see practicals/README.md)

1. Advection-diffusion-reaction equation in 1D with the Finite Difference (FD) method

The advection-diffusion-reaction equation is solved on a 1D domain using the finite-difference method. Constant, uniform velocity and diffusion coefficients are assumed. Spatial derivatives are discretized using 2nd-order, centered schemes. Time integration is carried out with different methods. Application to a metallic bar with fixed temperatures at the boundaries and possible heat exchange with the external environment.

• Time integration via ode45 solver: bar_1d_ode45.m
• Time integration via explicit (forward) Euler method: bar_1d_explicit_euler.m
• Time integration via implicit (backward) Euler method: bar_1d_implicit_euler.m

2. Advection-diffusion equation in 1D with the Finite Difference (FD) method

The advection-diffusion equation is solved on a 1D domain using the finite-difference method. Constant, uniform velocity and diffusion coefficients are assumed. The forward (or explicit) Euler method is adopted for the time discretization, while spatial derivatives are discretized using 2nd-order, centered schemes.

• Matlab script: advection_diffusion_1d.m
• Matlab live script: advection_diffusion_1d_live.mlx

3. Advection-diffusion-reaction equation in 1D with the Finite Volume (FV) method

The advection-diffusion-reaction equation is solved on a 1D domain using the finite-volume method. Constant, uniform velocity and diffusion coefficients are assumed. Spatial derivatives are discretized using 2nd-order, centered schemes. Time integration is carried out with ode15s solver. Application to a metallic bar with fixed temperatures at the boundaries and possible heat exchange with the external environment.

• Time integration via ode15s solver: bar_1d_ode15s.m

4. Advection-diffusion-reaction equation in 2D with the Finite Difference (FD) method

The advection-diffusion-reaction equation is solved on a 2D rectangular domain using the finite-difference method. Analyically prescribed velocity fields are assumed. Constant and uniform diffusion coefficients are assumed. Spatial derivatives are discretized using 2nd-order, centered schemes.

• Evaporating pool: evaporating_pool_2d_ode15s.m
• Tubular reactor: tubular_reactor_2d_ode15s.m
• Tubular reactor (multiple reactions): tubular_reactor_2d_multiple_reactions_ode15s.m

5. Advection-diffusion equation in 2D with the Finite Difference (FD) method

The advection-diffusion equation is solved on a 2D rectangular domain using the finite-difference method. Constant, uniform velocity components and diffusion coefficients are assumed. The forward (or explicit) Euler method is adopted for the time discretization, while spatial derivatives are discretized using 2nd-order, centered schemes.

• Matlab script: advection_diffusion_2d.m
• Matlab live script: advection_diffusion_2d_live.mlx

6. Poisson equation in 2D

The Poisson equation is solved on a 2D rectangular domain using the finite-difference method. A constant source term is initially adopted. Spatial derivatives are discretized using 2nd-order, centered schemes. Different methods are adopted for solving the equation: the Jacobi method, the Gauss-Siedler method, and the Successive Over-Relaxation (SOR) method

• Matlab script: poisson_2d.m
• Matlab live script: poisson_2d_live.mlx

The same Poisson equation is solved by explicitly assembling the sparse matrix corresponding to the linear system arising after the spatial discretization

• Matlab script: poisson_2d_matrix.m

7. Lorenz equations

The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. See also: Lorenz system

• Matlab script: lorenz_equations.m

8. Population Balance Equations (PBE)

Numerical solution of a diffusion controlled growth 1D population balance equation using the Discrete Sectional Method (DSM) or the Quadrature Method of Moments (QMOM).

• Matlab script (DSM): dsm_diffusion_controlled_growth.m
• Matlab script (QMOM): qmom_diffusion_controlled_growth.m
• Matlab live script (DSM): dsm_diffusion_controlled_growth_live.mlx
• Matlab live script (QMOM): qmom_diffusion_controlled_growth_live.mlx