Heat and Mass Transfer Project 4: 2D Transient Heat Transfer Analysis using Explicit Methods

Transient Thermal Analysis

Project Overview

This project explores numerical solutions to time-dependent (transient) heat conduction problems in two-dimensional rectangular domains using explicit finite difference methods. Students will develop a computational tool that simulates the dynamic temperature evolution within materials in response to changing boundary conditions and internal heat generation. This project bridges theoretical heat transfer principles with practical numerical methods essential for analyzing thermal systems under non-steady conditions.

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 an explicit finite difference solution to this equation.

Project Stages

1. Theoretical Foundation (25%)

2. Computational Implementation (40%)

Develop a program to implement the explicit finite difference method 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)
    • Stability parameters and output control options

  2. Spatial and Temporal Discretization:

    • Create a mesh for the rectangular domain with specified dimensions $W \times H$
    • Implement time stepping using the explicit scheme
    • Calculate and enforce the stability criterion
    • Handle boundary nodes appropriately for different boundary conditions

  3. Algorithm Implementation:

    • Develop the explicit time-marching algorithm
    • Implement heat generation terms in the domain
    • Apply boundary conditions at each time step:
      • Constant temperature (Dirichlet)
      • Constant heat flux (Neumann)
      • Convection (Robin): $-k\frac{\partial T}{\partial n} = h(T - T_{\infty})$
    • Track temperature evolution throughout the domain

  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

3. Validation and Verification (20%)

4. Case Studies and Analysis (15%)

Technical Requirements

Deliverables

  1. Comprehensive report including:

    • Theoretical background on explicit finite difference methods
    • Detailed mathematical formulation and stability analysis
    • Description of numerical implementation and algorithm flowchart
    • Validation results and verification approach
    • Case study description, results, and engineering interpretation
    • Limitations and potential improvements

  2. Well-documented source code:

    • Core program for solving 2D transient heat conduction
    • 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 the numerical approach
    • Demonstration of program capabilities
    • Presentation of key results and temporal evolution
    • Discussion of challenges and solutions

Evaluation Criteria

Bonus Challenges (Optional)

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

  1. Variable Thermal Properties

    • Implement temperature-dependent thermal properties (k, ρ, cp)
    • Analyze how variable properties affect stability criteria
    • Compare solutions with constant and variable properties

  2. Advanced Boundary Conditions

    • Implement time-dependent boundary conditions
    • Add radiation boundary conditions: $-k\frac{\partial T}{\partial n} = \epsilon\sigma(T^4 - T_{\infty}^4)$
    • Develop mixed boundary conditions (combinations of different types)

  3. Phase Change Implementation

    • Extend the code to handle phase change problems (melting/solidification)
    • Implement the effective heat capacity or enthalpy method
    • Track the movement of phase boundaries over time

  4. User Interface and Visualization

    • Create an interactive GUI for problem setup and simulation control
    • Develop advanced visualization tools for time-evolving thermal fields
    • Implement animation capabilities for temperature evolution
    • Create tools for parametric studies of transient responses
  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