Preface
This section provides some preliminary information about heat transfer equation, also known as the diffusion equation.
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Introduction to Linear Algebra with Mathematica
Glossary
Heat Conduction Equations
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The laws of thermodynamics tell us that no matter what the temperature distribution of the object is initially, the system must undergo a process that brings the object to thermal equilibrium. This process must obey the heat equation. This chapter deals with heat transfer processes that occur in solif matters without bulk motion of the matter. A solid (a block of metal, say) has one surface at a high temperature and one at a lower temperature. This type of heat conduction can occur, for example,through a turbine blade in a jet engine. The outside surface, which is exposed to gases from the combustor, is at a higher temperature than the inside surface, which has cooling air next to it. The level of the wall temperature is critical for a turbine blade.
The concept of heat flux was a key contribution of Joseph Fourier, in the analysis of heat transfer phenomena. According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. So flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time]^{−1}·[area]^{−1}. The area is of the surface the property is flowing "through" or "across". For example, the magnitude of a river's current, i.e. the amount of water that flows through a cross-section of the river each second, or the amount of sunlight energy that lands on a patch of ground each second, are kinds of flux. We denote the heat flux as
One dimensional Heat Transfer Equation in infinite strip
The one dimensional heat conduction equation
Metals | Ag | Cu | Al | Fe | Steel |
---|---|---|---|---|---|
κ [W/m-K] | 420 | 390 | 200 | 70 | 50 |
Non-metals | H_{2}O | Air | Engine oil | H_{2} | Brick | Wood | Cork |
---|---|---|---|---|---|---|---|
κ [W/m-K] | 0.6 | 0.026 | 0.15 | 0.18 | 0.6 | 0.2--0.5 | 0.04 |
To make the solution more meaningful and simpler, we group as many physical constants together as possible. Therefore, it is convenient to introduce dimensionless variables.
Convective heat transfer occurs whenever an object is either hotter or colder than a surrounding gas or fluid. Convection is a process of heat transfer between a surface and a fluid flowing in contact with it, never in a solid. It occurs always within a thin stagnant fluid film layer on the heat transfer surface. Heat transfer by convection is more difficult to analyze than heat transfer by conduction because no single property of the heat transfer medium, such as thermal conductivity, can be defined to describe the mechanism. Convective heat transfer is complicated by the fact that it involves fluid motion as well as heat conduction. Heat transfer by convection varies from situation to situation (upon the fluid flow conditions), and it is frequently coupled with the mode of fluid flow.
The primary equation for the rate of convection heat transfer is known as Newton's Law of Cooling:
There are two types of convection: natural convection and forced convection. Natural convection is produced by density differences in a fluid due to temperature differences (e.g., as in “hot air rises”). Global atmospheric circulation and local weather phenomena (including wind) are due to convective heat transfer. In forced convection, the fluid motion is generated by a source like a pump or a fan. It is one of the main forms of heat transfer used by engineers, because large amounts of thermal energy can be transported efficiently. It is used in heating and air conditioning systems, electronics cooling, and in numerous other technologies. Forced convection is used in designing heat exchangers, in which one fluid stream is used to heat or cool another fluid stream.
One of the most common examples of convection is natural convection in a non-mechanically ventilation/air conditioned building. As people enter a building, the lights get turned on and the sun heats the building, the air in the building begins to get warmer. The warm air is less dense than the air around it and begins to rise up and out of the building. The empty space left by the warm air is then replaced by cooler outside air and the cycle continues. This convective heat transfer through the movement of air is called natural convection. It is referred to as natural because it does not rely on a mechanical source like a fan to move the air.
Let u(x, t) denote the solution of the heat equation subject to the initial condition:
One dimensional Heat Equation for a finite rod
3D heat equation ??? to be fixed
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