Heat and Mass Transfer Project 5: 2D Transient Heat Transfer Analysis using Implicit Methods

Implicit Transient Thermal Analysis

Project Overview

This project explores numerical solutions to time-dependent (transient) heat conduction problems in two-dimensional rectangular domains using implicit finite difference methods. Students will develop a computational tool that simulates the dynamic temperature evolution within materials with the advantage of unconditional stability, allowing for larger time steps than explicit methods. This project connects theoretical heat transfer principles with advanced numerical methods used in engineering simulations where efficiency and stability are critical considerations.

Learning Objectives

Project Description

Background

The transient heat conduction in a 2D domain is governed by the following partial differential equation:

$$ \rho c_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k \frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k \frac{\partial T}{\partial y}\right) + \dot{q} $$

Where:

For homogeneous materials with constant thermal properties, this can be rewritten using thermal diffusivity $\alpha = k/(\rho c_p)$:

$$ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) + \frac{\dot{q}}{\rho c_p} $$

This project focuses on developing implicit finite difference solutions to this equation, which evaluate spatial derivatives at the future time step.

Project Stages

1. Theoretical Foundation (25%)

2. Computational Implementation (40%)

Develop a program to implement implicit finite difference methods for 2D transient heat conduction. Tasks include:

  1. Program Input Requirements:

    • Material properties (thermal conductivity, density, specific heat)
    • Internal heat generation rates (constant or time-dependent)
    • Grid size (number of nodes in x and y directions)
    • Time step and total simulation time
    • Initial temperature distribution
    • Boundary condition specifications for each edge (type and values)
    • Solution method parameters (direct or iterative solver options)

  2. Spatial and Temporal Discretization:

    • Create a mesh for the rectangular domain with specified dimensions $W \times H$
    • Implement time stepping using implicit formulations
    • Handle boundary nodes appropriately in the coefficient matrix

  3. Matrix Assembly and Solution:

    • Generate the coefficient matrix for the system of equations at each time step
    • Implement boundary conditions into the matrix structure:
      • Constant temperature (Dirichlet)
      • Constant heat flux (Neumann)
      • Convection (Robin): $-k\frac{\partial T}{\partial n} = h(T - T_{\infty})$
    • Create the right-hand side vector incorporating the previous time step solution
    • Implement efficient solution methods

  4. Program Output Requirements:

    • Temperature distribution at specified time intervals
    • Temperature history at user-defined monitoring points
    • Heat fluxes across the domain at selected times
    • Visualization of temperature evolution through animations or time sequence plots
    • Energy balance verification at each time step
    • Performance metrics (computation time, memory usage)

3. Validation and Verification (20%)

4. Case Studies and Analysis (15%)

Technical Requirements

Deliverables

  1. Comprehensive report including:

    • Theoretical background on implicit finite difference methods
    • Detailed mathematical formulation and stability analysis
    • Description of numerical implementation with focus on coefficient matrix assembly
    • Explanation of system solution approaches
    • Validation results and verification approach
    • Case study description, results, and engineering interpretation
    • Computational performance analysis
    • Limitations and potential improvements

  2. Well-documented source code:

    • Core program for solving 2D transient heat conduction
    • Matrix assembly and solution routines
    • Supporting functions for time stepping, visualization, etc.
    • User manual explaining how to use the program
    • Example input files for validation cases

  3. Presentation:

    • Clear explanation of implicit finite difference method advantages
    • Demonstration of program capabilities
    • Presentation of key results and computational performance
    • Discussion of challenges and solutions

Evaluation Criteria

Bonus Challenges (Optional)

For students interested in extending their understanding and skills beyond the core requirements:

  1. Advanced Implicit Schemes

    • Implement the Alternating Direction Implicit (ADI) method
    • Compare computational efficiency between different implicit approaches

  2. Adaptive Time Stepping

    • Implement variable time step control based on error estimates
    • Develop strategies for automatic time step adjustment
    • Analyze the efficiency gains from adaptive versus fixed time stepping

  3. Non-Linear Thermal Problems

    • Extend the code to handle temperature-dependent material properties
    • Implement radiation boundary conditions: $-k\frac{\partial T}{\partial n} = \epsilon\sigma(T^4 - T_{\infty}^4)$
    • Develop linearization techniques and iterative approaches for non-linear terms

  4. Multi-Domain Problems

    • Implement capabilities for handling multiple materials with different properties
    • Develop interface conditions between different materials
    • Analyze the impact of thermal contact resistance at material interfaces
  1. Fundamentals of Heat and Mass Transfer, [8th ed.] by Theodore L. Bergman, Adrienne S. Lavine
  2. Heat and Mass Transfer, Fundamentals & Applications, [6th ed.] by Yunus A. Cengel, Afshin J. Ghajar
  3. Heat Conduction, [5th ed.] by Kakac, S., Yener, Y., & Naveira-Cotta, C.P.
  4. Heat Conduction, [3rd ed.] by David W. Hahn, M. Necati Özisik
  5. Finite Difference Methods in Heat Transfer, [2nd ed.] by Özişik, M.N., Orlande, H.R.B., Colaço, M.J., & Cotta, R.M.

Submission Guidelines