What is Heat diffusion: Definition and 25 Discussions

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003. Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the heat equation have also been given much attention in the numerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

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  1. H

    Find equilibrium profile T(x) Between Two Rods

    Knowing that we are in equilibrium ##\frac{\partial}{\partial t} = 0##. We now have a Laplace's equation ##\kappa \frac{\partial^2 T}{\partial x^2} = 0## I separated the rod in 2 halves. The solution of this equation is ##\kappa_1 \frac{\partial2 T}{\partial x2} = C_1## Integrating both side...
  2. user123abc

    Model CO2 diffusing across the wall of a cylindrical alveolar blood vessel

    TL;DR Summary: Solve heat equation in a disc using fourier transforms Carbon dioxide dissolves in the blood plasma but is not absorbed by red blood cells. As the blood returns to an alveolus, assume that it is well-mixed so that the concentration of dissolved CO2 is uniform across a...
  3. majorjc

    I Heat diffusion through grain mash to reach equilibrium

    I was reading posts this morning on another forum and came across a question that made me start thinking about physics. Since I've always tried to satisfy my curiosity through reading and trying to learn new things, or trying to remember the things that I've forgotten years ago, I went to...
  4. P

    Heat Diffusion Equation - Using BCs to model as an orthonormal system

    I've tried to show b) by using the sine Fourier series on ##[0,2a]##, to get ##g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)## Therefore ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx## These are equal therefore it is an orthonomal basis. I'm not sure if this is correct so...
  5. Vick

    I Alternating Direction Implicit method for solving 2D Heat diffusion

    I'm trying to compute a 2D Heat diffusion parabolic PDE: $$ \frac{\partial u}{\partial t} = \alpha \{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \} $$ by the ADI method. I am actually trying to go over the example in this youtube video. The video is in another...
  6. K

    Heat diffusion into an infinite rod

    I have recently been curious about heat diffusion. If there is space in one dimension with any kind of temperature dispersed throughout, then the heat equation states that the derivative of the temperature with respect to time at any point equals some constant (k) multiplied by the second...
  7. J

    Canning/Heat-preserving food

    Preserving food at home involves putting produce in glass jars and heating them to kill pathogens like bacteria, fungus, yeast, etc... The USDA recommends AGAINST using an oven to preserve food. They only support the boiling water method, which involves submerging the jars in a pot of boiling...
  8. A

    Modeling analytical solution of 1D heat equation

    I am trying to write code for analytical solution of 1D heat conduction equation in semi-infinite rod. The analytical solution is given by Carslaw and Jaeger 1959 (p305) as $$ h(x,t) = \Delta H .erfc( \frac{x}{2 \sqrt[]{vt} } ) $$ where x is distance, v is diffusivity (material property) and t...
  9. A

    A How to validate a code written for solution of 1D diffusion?

    Consider the conceptual model presented in the attached image, of heat conduction in a bar. There is a heat source at left side and heat is observed at point Ho after a distance L from the source. If we consider only heat transfer through conduction then this problem can be modeled by...
  10. H

    Interesting Heat Diffusion Problem

    We solved a simple looking heat diffusion problem because it describes an apparatus used in an NIH sponsored research project. It resembles problems discussed in textbooks and in many papers on the web. A textbook method solves all those problems but a straight forward application of that...
  11. T

    Heat diffusion in a spherical shell

    Hey guys, I have a problem that is giving me trouble. Homework Statement I have to solve time dependent diffusion equation ##D\nabla^2 T(r,t)=\frac{\partial T}{\partial t}## (##D## is diffusion constant and ##T(r,t)## is temperature function) for a spherical shell of radii ##r_1## and ##r_2##...
  12. M

    Anisotropic heat diffusion in gas streams with shear?

    Hi all, I'm working on a heat transfer problem with a gas stream in a tiny tube. At my dimensions and flow rates, the flow still has a parabolic velocity profile. The mean radial velocity of the gas is zero, and I've treated the radial aspect of the heat transfer as strictly diffusion...
  13. L

    Numerical method for a heat diffusion problem

    Hello everyone! I hope you may be able to shed light on a recent problem that i have to deal with. I keep working on a numerical solution on a heat diffusion problem. It is a numerical solution for the PDE θT/θt=a*(θ^2T/θx^2 + θ^2Τ/θy^2), that describes the heat diffusion on a two...
  14. Z

    PDE for Heat Diffusion Equation

    Homework Statement The one-dimensional heat diffusion equation is given by : ∂t(x,t)/∂t = α[∂^2T(x,t) / ∂x^2] where α is positive. Is the following a possible solution? Assume that the constants a and b can take any positive value. T(x,t) = exp(at)cos(bx) Homework Equations...
  15. X

    Problem solving Heat Diffusion Equation

    Homework Statement One face of a thick uniform layer is subject to a sinusoidal temperature variation of angular frequency ω. SHow that the damped sinusoidal temperature oscillation propagate into eh layer and give an expression for the decay length of the oscillation amplitude. A cellar...
  16. L

    Numerical method on heat diffusion problem

    Hello there! I am solving a heat diffusion problem which is described by the differential equation: θΤ/θt=a*(θ^2Τ/ΘΤ^2). I will apply two diffrent temperatures on the ends of a metal rod. Solving the equation i conclude, according to the boundary conditions of course, at the function...
  17. F

    2D heat diffusion simulation with thermal insulators

    Hi. I don't know if this is the correct place to ask this, but prefer you to suggest me where I should ask. I'm starting with numerical simulation and I've been playing with the finite differences method to solve the heat equation on 1D, 2D and 3D uniform grids. This was really simple. Now I...
  18. H

    Heat diffusion equation solutions for semi-infinite slab

    Homework Statement http://img42.imageshack.us/img42/1082/clipboard01lx.jpg [Broken] Homework Equations (see solution) The Attempt at a Solution I literary just spent 5 hours trying to apply those boundary conditions, trying exponentials, sines, cosines, hyperbolic function etc...
  19. L

    Solving 2D Heat Diffusion Eqn w/ Finite Difference Method

    I have to write a FD expilicit method, for temp dist on 2D plane. I am trying to mod the 1D solver to 2D solver, the code below is a 1D solver. Any possible suggestions how it can be done? I initially, derived u(i,j+1) for 2D for in that derivation i,j corresponds to x,y... where as in 1D...
  20. S

    Fourier method for non-uniform heat diffusion

    Homework Statement The Fourier method specifies that change of heat energy of = heat out from left boundary - heat in from right boundary segment in time Δt If vars such as c,p.. are constant in the equation, and T(-L,L) = T_a, does the the aforementioned Fourier method...
  21. S

    Heat diffusion with arrhenius source

    Homework Statement a material occupies -L < x < L and has uniform ambient temperature T_a. A chemical reaction begins within the body leading to the 1-d heat equation: pc \frac {\partial{T}}{\partial{t}} = k \frac {\partial^2{T}}{\partial{x^2}} + pQAe^\frac{-E}{RT} with BC and IC T(+/-...
  22. S

    Heat diffusion with source

    Homework Statement a material occupies -L < x < L and has uniform ambient temperature T_a. A chemical reaction begins within the body leading to the 1-d heat equation: pc \frac {\partial{T}}{\partial{t}} = k \frac {\partial^2{T}}{\partial{x^2}} + pQAe^\frac{-E}{RT} with BC and IC T(+/-...
  23. B

    Verifying 1D Heat Diffusion PDE Solutions

    I have trying solving this PDE for some random boundary values, and were wondering if someone could verify my calculations? \[ \begin{array}{l} T_t = DT_{xx} \\ T\left( {0,t} \right) = 0,T(\pi ,t) = 0,T(x,0) = \frac{1}{4}\left( {\left( {x - \frac{\pi }{2}} \right)^2 + \frac{{\pi ^2...
  24. C

    Sneaker heat diffusion through walking

    Hi, I'd like to figure out the amount of heat lost in the kinematic action of putting your foot on the ground while walking. I'm not interested in the diffussion of body heat through the shoe or anything, just the heat energy released by the impact itself. I know this is obscure, and probably...
  25. N

    Heat diffusion throughout a sphere

    Hey there I was wondering if someone maybe be able to help me with a couple of questions I have. I attach a set of equations the help define the heat diffusion throughout a sphere could you please tell me; in V=rT What is V when r is the radius of the sphere and T is the temperature ...
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