Heat Transfer

Advanced Heat Transfer

Conduction: Analytical methods in heat conduction. Numerical methods for steady and unsteady state conduction problems.

Analytical Techniques to Solve Conduction Problem

2D Steady State Conduction Example of Rectangular Plate

The linear homogeneous second-order differential equation in two dimensions, commonly known as the Laplace equation, is analytically solved to model steady-state heat conduction without internal heat generation. In this example, three of the four boundary conditions are homogeneous, and temperature normalization is applied using the Max-Min Scaling method.

Derivation of Temperature Distribution in Unsteady Heat Conduction

The temperature distribution in unsteady heat conduction is derived as a function of one spatial dimension (x) and time (t). The analytical solution is obtained by applying boundary and initial conditions for a slab with convective boundaries. After solving the transcendental equation and using orthogonal functions to determine constants, the first-term approximation is considered valid for Fourier numbers greater than 0.2. Heisler charts are also applicable for different geometries: (i) slab, (ii) cylinder, and (iii) sphere.

Finally, the lumped capacitance method is derived, where temperature is only a function of time.

Derivation of Temperature Distribution in Early Regime (Fourier number < 0.05) of Unsteady Heat Conduction

This study examines the early regime of unsteady heat conduction in a semi-infinite solid, where the short process time allows for an infinite length scale assumption. The temperature distribution T(x,t), varying in one spatial dimension and time, is derived by reducing the system to a single variable through similarity transformation and order of magnitude analysis. The derivation employs applied mathematics techniques, notably the error function and gamma function.

First Term Approximation Using Transcendental Equation.

Heisler chart for center-line temperature and temperature at a location.

Numerical methods to Solve Conduction Problem

Numerical Heat Transfer

Conduction: Discretization methods; Control Volume Formulation; Numerical solution of steady and un-steady conduction; 1D and 2D problems; solution of algebraic equation; relaxation technique.

2D Transient Conduction - Finite Volume Explicit Method
Dirichlet Boundary Condition with no Heat Source

A MATLAB tutorial focused on two-dimensional transient conduction using the Finite Volume Explicit Method. It demonstrates how to code a MATLAB simulation for heat diffusion in a rectangular plate, addressing specific boundary conditions. This tutorial is particularly useful for understanding the numerical approach to solving heat transfer problems and can serve as an educational resource for those learning or applying similar methods in thermal analysis.

Source Code:

https://github.com/shehzaibyk/

2D Transient Conduction - Finite Volume Implicit Method
Dirichlet Boundary Condition with no Heat Source

A MATLAB tutorial on solving 2D transient heat conduction using the Finite Volume Method with an implicit approach and Gauss-Jacobi iterations. It guides viewers through coding the time-dependent heat diffusion in a rectangular plate, considering specific boundary conditions and initial setup.

Page updated

Google Sites

Report abuse