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Free PDF Download: Conduction of Heat in Solids by H.S. Carslaw and J.C. Jaeger


Conduction of Heat in Solids: A Comprehensive Guide




Conduction of heat in solids is one of the most important topics in thermal engineering. It deals with how heat is transferred within and between solid materials due to the difference in temperature. Understanding conduction of heat in solids is essential for designing and optimizing various engineering and industrial processes, as well as for fire protection and safety, and biological and medical systems. In this article, we will provide a comprehensive guide on conduction of heat in solids, covering the following aspects:


  • What is conduction of heat in solids?



  • How to solve conduction problems?



  • What are the applications of conduction of heat in solids?



By the end of this article, you will have a clear and thorough understanding of conduction of heat in solids, and you will be able to apply it to your own projects and problems.




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What is Conduction of Heat in Solids?




Conduction of heat in solids is the process of heat transfer within or between solid materials due to the difference in temperature. Heat is a form of energy that flows from a region of higher temperature to a region of lower temperature. Conduction is one of the three modes of heat transfer, along with convection and radiation.


Conduction of heat in solids only occurs in a medium, which can be a gas, liquid, or solid. The medium must be stationary, meaning that there is no bulk motion or flow. If there is a mass-averaged velocity, then the heat transfer is called convection. Conduction is also different from radiation, which does not require a medium and can occur in a vacuum.


The Mechanism of Conduction




The mechanism of conduction depends on the type and structure of the solid material. In general, there are two main mechanisms that allow heat to be conducted in solids:


  • Molecular vibration: This mechanism occurs in all solids, but it is more dominant in non-metallic solids. In this mechanism, the atoms or molecules in the solid vibrate around their equilibrium positions due to thermal energy. When they collide with each other, they transfer some of their kinetic energy to their neighbors. This results in a net flow of heat from the hotter regions to the colder regions.



  • Free electron movement: This mechanism occurs mainly in metallic solids, which have a large number of free electrons that can move freely throughout the solid. In this mechanism, the free electrons carry thermal energy as they move randomly under the influence of an electric field or a temperature gradient. They collide with each other and with the atoms or ions in the solid, transferring some of their kinetic energy to them. This results in a net flow of heat from the hotter regions to the colder regions.



The Factors Affecting Conduction




The rate or amount of conduction of heat in solids depends on several factors, such as:


Temperature Gradient




The temperature gradient is the change in temperature per unit distance along a given direction. It indicates how steeply the temperature varies within the solid. The higher the temperature gradient, the faster the heat transfer by conduction.


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Thermal Conductivity




The thermal conductivity is a property of the material that measures how well it conducts heat. It depends on the type, structure, and composition of the material, as well as on the temperature and pressure. The higher the thermal conductivity of the material, the faster the heat transfer by conduction.


Cross-Sectional Area




The cross-sectional area is the area of the solid that is perpendicular to the direction of heat transfer. It determines how much heat can flow through the solid at a given time. The larger the cross-sectional area, the faster the heat transfer by conduction.


Thickness




The thickness is the distance between the two surfaces of the solid that are at different temperatures. It indicates how far the heat has to travel within the solid. The smaller the thickness, the faster the heat transfer by conduction.


How to Solve Conduction Problems?




Conduction problems are mathematical problems that involve finding the temperature distribution and/or the heat flux within or between solid materials due to conduction. To solve conduction problems, we need to use the following steps:


  • Identify the system and its boundaries: The system is the solid or a part of it that we are interested in analyzing. The boundaries are the surfaces or interfaces that separate the system from its surroundings or other materials.



  • Apply the general heat conduction equation: The general heat conduction equation is a partial differential equation that describes how the temperature varies within the system due to conduction. It can be written as: $$\frac\partial\partial t(\rho c_p T) = \nabla \cdot (k \nabla T) + q'''$$ where $\rho$ is the density, $c_p$ is the specific heat, $T$ is the temperature, $t$ is the time, $k$ is the thermal conductivity, $\nabla$ is the gradient operator, and $q'''$ is the volumetric heat generation rate.



Apply the boundary conditions: The boundary conditions are equations or expressions that specify the temperature or the heat flux at the boundaries of the system. They can be of three types:


  • Dirichlet boundary condition: The temperature at a boundary is given or known.



  • Neumann boundary condition: The heat flux at a boundary is given or known.



  • Robin boundary condition: The heat flux at a boundary is proportional to the difference between the temperature at the boundary and a reference temperature.



  • Apply the initial condition: The initial condition is an equation or expression that specifies the temperature distribution within the system at a given initial time.



  • Solve for the unknowns: The unknowns are usually the temperature distribution and/or the heat flux within or between solid materials. Depending on the complexity of the problem, we can use different methods of solution, such as analytical methods or numerical methods.



The General Heat Conduction Equation




The general heat conduction equation is derived from applying the conservation of energy principle to a differential control volume within the solid. It states that the rate of change of thermal energy within the control volume is equal to the net rate of heat transfer by conduction across the boundaries of the control volume plus the rate of heat generation within the control volume. The general heat conduction equation can be simplified or modified depending on the assumptions and conditions of the problem, such as steady or transient state, one-dimensional or multi-dimensional, isotropic or anisotropic, homogeneous or heterogeneous, etc.


The Boundary Conditions




The boundary conditions are essential for solv


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