This app solves the heat equation in 2-dimensions given initial and boundary conditions and plots the results.

x_0_bc

x_1_bc

y_1_bc

<h3>Boundary Conditions</h3>Boundary conditions must be set for each of the 4 edges of the domain. Either set temperature, Dirichlet, or set temperature flux, Neumann, boundary conditions can be set here.

<h3>Simulation Time</h3>Set the total time that the simulation will be run for as well as the size of the time step.A time step must be selected so that {{ "time\\ step \\leq \\frac{(spatial\\ step)^2}{4 \\alpha}" | katex }} to ensure that the FTCS method used to calculate the solution will be stable.Since each step in the x and y directions is linked to the number of pixels in the image provided above, {{ "spatial\\ step = pixel\\ to\\ distance\\ conversion\\ factor" | katex }}.

x_0_bcselection

The number of output plots can be set below. Plots will be evenly spaced over the total time set above.

plotnum

x_1_bcselection

y_0_bcselection

y_1_bcselection

maxtemp

time

mintemp

<h3>Thermal Diffusivity</h3>Thermal diffusivity is a value that measures the rate of heat transfer in a material. Examples of thermal diffusivity for various materials can be found here: <a href="https://en.wikipedia.org/wiki/Thermal_diffusivity" rel="noopener noreferrer" target="_blank">https://en.wikipedia.org/wiki/Thermal_diffusivity</a>

An example image is provided to the right, a screenshot of it can be used as the image input.

<h3>Domain Size and Initial Condition</h3>Both the size of the spatial domain and the initial condition (temperature distribution at {{ "t=0\\, s" | katex }}) are set using a .png or.jpg image file. The size of the file in pixels sets the domain, where each pixel equals one step in the horizontal (x) or vertical (y) direction. The physical distance that each pixel represents in {{ "cm" | katex }} can be set by "pixel to distance conversion factor" below. When the result is calculated, each pixel in the image will represent a number of "steps" in the x or y direction equal to this conversion factor. The initial condition, or temperature distribution at {{ "t=0\\, s" | katex }}, is set by how dark each pixel in the image is. Completely black pixels are set to the maximum temperature and completely white pixels to the minimum temperature, both set below.

alpha

pixel_2_cm

timestep

<h2>Solving the unsteady heat equation in 2-dimensions</h2> The heat equation in 2-dimensions is a partial differential equation that is a function of time and two spatial dimensions. By setting an initial condition and 4 boundary conditions, one for each side of the 2D domain, the temperature distribution over the domain can be solved for. This app obtains a numerical solution to the 2D heat equation using method known as "Forward Time Centered Space" or FTCS. <h3>Heat Equation in 2D</h3> The heat equaion in 2D can be written as: {{ "\\frac{\\partial T}{\\partial t} = \\alpha \\left[ \\frac{\\partial^2 T}{\\partial x^2} + \\frac{\\partial^2 T}{\\partial y^2} \\right]" | katex }}where:{{ "T = " | katex }} temperature {{ "[^{\\circ}C]" | katex }}{{ "t = " | katex }} time {{ "[s]" | katex }}{{ "\\alpha = " | katex }} thermal diffusivity coefficient {{ "[cm^2/s]" | katex }}﻿{{ "x = " | katex }} width {{ "[cm]" | katex }}﻿{{ "y = " | katex }} length {{ "[cm]" | katex }}

2D Heat Equation Visualizer