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Topic: Heat equation


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  Heat equation - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-06)
Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object.
Heat flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.
The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options.
en.wikipedia.org /wiki/Heat_equation   (2009 words)

  
 Heat - Wikipedia, the free encyclopedia
The amount of heat exchanged during a phase change is known as latent heat and depends primarily on the substance and the initial and final phase.
Heat is a process quantity—as opposed to being a state quantity—and is to thermal energy as work is to mechanical energy.
Heat flows between regions that are not in thermal equilibrium with each other; it spontaneously flows from areas of high temperature to areas of low temperature.
en.wikipedia.org /wiki/Heat   (2465 words)

  
 Heat conduction - Wikipedia, the free encyclopedia
Heat transfer is always directed from a higher to a lower temperature.
R-value is the unit for heat resistance, the reciprocal of the conductance.
When heat is being conducted from one fluid to another through a barrier, it is sometimes important to consider the conductance of the thin film of fluid which remains stationary next to the barrier.
en.wikipedia.org /wiki/Heat_conduction   (443 words)

  
 Heat equation: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-11-06)
In mathematics, and in particular analysis, a partial differential equation (pde) is an equation involving partial derivatives of an unknown function....
Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat heat quick summary:
Heat (abbreviated q, also called heat change) is the transfer of thermal energy between two bodies which are at different temperatures....
www.absoluteastronomy.com /encyclopedia/h/he/heat_equation.htm   (1201 words)

  
 Heat Conduction
and the heat flow are, in fact, oppositely directed, for the gradient (by definition) is in the direction of increasing temperature, while heat flows from higher to lower temperatures.
, and that the heat flux is proportional to the magnitude of the gradient.
Equation (2.1), the fundamental relation in heat conduction, is called Fourier's law.
www.ibiblio.org /links/devmodules/heatflow/compat/page12.html   (192 words)

  
 Heat conduction of a moving heat source:
In fact, since heat is not a vector quantity, the effects of heat input at different times can be added directly.
So for a time dependent heat source, the temperature distribution at any time is the integration of previous temperatures.
Heat conduction of a moving heat source is of interest because in laser cutting and scribing laser beam is in relative movement to the part.
www.columbia.edu /cu/mechanical/mrl/ntm/level2/ch03/html/l2c03s05.html   (969 words)

  
 What is the one-dimensional heat equation?
The heat equation is a partial differential equation that describes the flow of heat in a material in which the rate of heat flow is proportional to the temperature gradient.
The heat equation is also called the diffusion equation, since the same equation can be used to describe the diffusion of quantities other than heat.
The one-dimensional heat equation describes the flow of heat in a one-dimensional system, such as in a wire.
support.wolfram.com /mathematica/mathematics/numerics/pde1.html   (472 words)

  
 One-Dimensional Heat Equation, Part 1
The heat equation models the flow of heat in a rod that is insulated everywhere except at the two ends.
The "one-dimensional" in the description of the differential equation refers to the fact that we are considering only one spatial dimension.
It depends on the thermal conductivity of the material composing the rod, the density of the rod, and the specific heat of the rod.
www.math.duke.edu /education/ccp/materials/engin/pdeintro/pde1.html   (662 words)

  
 NWS Birmingham, AL - Heat Index   (Site not responding. Last check: 2007-11-06)
Heat index or HI is sometimes referred to as the "apparent Temperature".
The equation was obtain by multiple regression analysis and there is a ±1.3 degree °F error.
The equation is only useful for temperatures 80 degrees or higher, and relative humidities 40% or greater.
www.srh.noaa.gov /bmx/tables/hindex.html   (217 words)

  
 Intercooler Theory   (Site not responding. Last check: 2007-11-06)
Heat is transferred from the hot tubes and fins to the cool outside air.
A is the heat transfer area, or the surface area of the intercooler tubes and fins that is exposed to the outside air.
Heat transfer goes really well when there is a large temperature difference, or driving force, between the two fluids.
www.gnttype.org /techarea/turbo/intercooler.html   (4249 words)

  
 Examples (Partial Differential Equation Toolbox)
The square region consists of a material with coefficient of heat conduction of 10 and a density of 2.
In the square region, enter a density of 2, a heat capacity of 0.1, and a coefficient of heat conduction of 10.
In the diamond-shaped region, enter a density of 1, a heat capacity of 0.1, and a coefficient of heat conduction of 2.
www.weizmann.ac.il /matlab/toolbox/pde/2examp20.html   (634 words)

  
 General Conduction Theory in Heat Transfer
This equation determines the heat flux vector q for a given temperature profile T and thermal conductivity k.
The temperature profile within a body depends upon the rate of its internally-generated heat, its capacity to store some of this heat, and its rate of thermal conduction to its boundaries (where the heat is transfered to the surrounding environment).
In the Heat Equation, the power generated per unit volume is expressed by q
www.efunda.com /formulae/heat_transfer/conduction/overview_cond.cfm   (232 words)

  
 The fundamental solution of the heat equation
Notice that the Gaussian distribution of the heat kernel becomes very narrow when t is small, while the height scales so that the integral of the distribution remains one.
Perhaps this is not too surprising when we think that the heat kernel itself solves the heat equation, and changes instantaneously from a delta function to a smooth, Gaussian distribution.
Show that if we assume that w depends only on r, the heat equation becomes an ordinary differential equation, and the heat kernel is a solution.
www.mathphysics.com /pde/ch20wr.html   (1107 words)

  
 The Heat Equation -- Building a Model   (Site not responding. Last check: 2007-11-06)
One side is often heated by boiling water, maintaining a constant temperature of 100 degrees Celsius (at sea level), and the other side is often immersed in a mixture of ice and water, maintaining a constant temperature of zero degrees Celsius.
Similarly, heat is flowing into the middle object at the left and out of the middle object at the right at the same rate.
The first term on the right in the first equation comes from the heat flow from the layer to the left and the second term comes from the heat flow from the layer to the right.
www.math.montana.edu /frankw/ccp/multiworld/partdes/heatbuilding/body.htm   (1622 words)

  
 Derivation of heat equation
the heat energy contained in a material is proportional to the temperature, the density of the material, and a physical characteristic of the material called the specific heat capacity.
In other words, the rate of heat flow from one region to another is proportional to the temperature gradient between the two regions.
You will probably also agree that the rate of heat flow will be proportional to the area of the contact; for example, a short pin with one end on a hot stove and the other touching your hand is preferable to putting the palm of your hand on a frying pan.
www.mathphysics.com /pde/HEderiv.html   (484 words)

  
 The heat equation and Fourier series   (Site not responding. Last check: 2007-11-06)
We will study the solution to the heat equation of a wire of length L. The body of the wire is insulated but the ends may or may not be (depending on the boundary conditions we shall set).
This is the partial diffrential equation governing the temperature function u(x,t).
The reason for using sine series to solve a zero ends heat equation is because the heat equation is much easier to solve when the initial temperature is a sine or a cosine function and the sine function matches the zero ends boundary conditions.
web.usna.navy.mil /~wdj/sm311o_heat1.htm   (225 words)

  
 CS267: Notes for Lecture 13, Feb 27, 1996
We will derive the heat equation in some detail, and later show that the computational bottleneck is identical to that for electrostatic or gravitational potential.
Dividing (H(x-h)-H(x)) by h yields the rate at which the "heat density" builds up at x, which is in turn proportional to the rate at which the temperature rises or falls at x.
Even on a serial machine, the linear system for one step of Crank-Nicholson on the 2D heat equation is a much more interesting linear system to solve than the 1D case, where we had a tridiagonal system.
www.cs.berkeley.edu /~demmel/cs267/lecture17/lecture17.html   (3566 words)

  
 THE HEAT EQUATION   (Site not responding. Last check: 2007-11-06)
The physical problem considered here concerns the flow of heat in a 2-dimensional domain represented by n x n cells.
The problem is presently set to have fix temperature (blue) on the boundary, with the exception of the random sources which may happen to be located there.
You should noticed that if no heat sources are generated (and maintained) on the boundary, the equilibrium temperature will be initial minimum temperature (blue).
www.math.utah.edu /~veronese/heat.html   (236 words)

  
 Physics of the Heat Equation
The way the heat flows across some domain and some dimension has been a field of physics that has had its start in the time of Newton.
Since then, with the help of partial differential equations and other techniques, it has been possible to model how heat flows by numerical means.
There are two states with which the heat equation is solved that we will consider; transient and steady state.
www.cs.toronto.edu /~arnold/492/ParallelAlgorithms/commentedReport/node4.html   (293 words)

  
 Modelling: Derivation of the heat equation
To derive the heat equation, we will consider the flow of heat along a metal rod.
The heat flow is proportional to the temperature gradient, i.e.
The heat equation has the same form as the equation describing diffusion.
www-solar.mcs.st-and.ac.uk /~alan/MT2003/PDE/node20.html   (139 words)

  
 Session 8 \\{\bf Multi-Dimensional, Steady-State, Linear Heat Conduction: Analytical Methods}
The steady state heat equation must be solved subject to appropriate boundary conditions.
The one dimensional forms of the steady state heat equation are particularly simple and readily solvable in many cases as they reduce to second order ordinary differential equations.
Consider steady state heat conduction in a thin rectangular plate of width 1 and height 1.
www.rh.edu /~ernesto/C_S2002/CHT/notes/s08/s08.html   (1571 words)

  
 Body
The first of equations (3) is Newton's law f=ma for a fluid element subject to the external force f and to forces of pressure and friction, and
Equations (3) and (4) apply in principle both to laminar and turbulent flows although, because of the impossibility of following all the minor fluctuations in velocity associated with turbulence and because of the difficulty and lack of solutions in turbulence problems, they cannot be used directly to solve problems in turbulent flow.
The justification for equations (5) assumes that u, the x component of velocity, is mainly a function of x so that its y and z derivatives tend to zero.
www.coolissues.com /mathematics/Navier-Stokes/nstokes.htm   (1457 words)

  
 The Navier-Stokes equation   (Site not responding. Last check: 2007-11-06)
in the internal energy equation corresponding to the generation of heat by viscous damping are also unimportant in most problems.
Hence the number of equations is the same as the number of independent dynamical variables.
One important example where they do not is the heating in an accretion disk where friction between adjacent annuli in the disk generates heat.
astron.berkeley.edu /~jrg/ay202/node50.html   (537 words)

  
 The Jeffreys Type Heat flux Model and Relevant Associated Equations   (Site not responding. Last check: 2007-11-06)
When Gurtin and Pipkin derived this equation it should be noted that it was under the assumption that the heat flux relaxation function is bounded and hence the equation disallowed a delta function in the kernel.
Note that equation (31) is the integral form of equation (32) to follow.
When the temperature is much higher than the Debye temperature, the U-processes dominate, and this equation reduces to the Cattaneo's hyperbolic heat conduction form.
www.msi.umn.edu /~xiangmin/paper/hhc97/node8.html   (808 words)

  
 The Heat Equation   (Site not responding. Last check: 2007-11-06)
In this example a=1, l=10 and the intial amplitude consists of one bump centered on x=5, just as in the wave equation demonstration.
The demonstration plots the solution given by separation of variables that you have found in class.
Contrast the time dependence of these coefficients with that of the corresponding coefficients for the wave equation.
www.math.ubc.ca /~feldman/demos/demo7.html   (129 words)

  
 Transport of Heat   (Site not responding. Last check: 2007-11-06)
Conservation of heat yields the transport equation for heat, or rather, for the change of temperature.
The equation is identical to the advection-reaction-dispersion equation for a chemical substance:
The similarity between thermal and hydrodynamic transport is an approximation which mainly falls short because diffusion of mass is by orders of magnitude larger in water than in minerals, whereas diffusion of heat is comparable in the two media although often anisotropic in minerals.
wwwbrr.cr.usgs.gov /projects/GWC_coupled/phreeqc/html/final-23.html   (325 words)

  
 Heat equation pictures for Math 421, fall 2004
When t gets large, the heat oozes out the ends, and the temperature drops to 0 in the whole interval.
When t gets large, the heat tends to even out over the whole interval, and the temperature approaches a constant (determined by the total area).
When t gets large, the heat oozes out the left end, while the temperature curve always has a horizontal tangent at the right end because there is no heat flow there.
www.math.rutgers.edu /~greenfie/mill_courses/math421a/heat_eq_pix.html   (199 words)

  
 [No title]   (Site not responding. Last check: 2007-11-06)
Nevertheless, these techniques are rather involved and their application to the actual heat calculation in a nuclear reactor are not so frequent.
For many reasons, it is preferred to perform the calculation of the heat conduction by numerical methods.
In conclusion we have obtained a set of N algebraic equations that we have to solve for the unknown temperature at the point i (i=1,2,...,N).
www.nuc.berkeley.edu /thyd/ne161/gregori/section7.html   (465 words)

  
 Solution of the heat equation: separation of variables
This is different from the wave equation where the oscillations simply continued for all time.
For problems involving heat flow there is no reason why the temperature should have the same value at each end of the rod.
satisfies the heat equation and the boundary conditions for the full problem.
www-solar.mcs.st-and.ac.uk /~alan/MT2003/PDE/node21.html   (423 words)

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