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The Heat Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi An Easy Solution to Industrial Heat Transfer Problems…
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The Heat Equation Incorporation of the constitutive equation into the energy equation above yields: Dividing both sides by r Cp and introducing the thermal diffusivity of the material given by
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For constant thermal properties and no heat generation. This is often called the heat equation.
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General conduction equation based on Cartesian Coordinates
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For an isotropic and homogeneous material:
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General conduction equation based on Polar Cylindrical Coordinates
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Thermal Conductivity of Brick Masonry Walls
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Thermally Heterogeneous Materials
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Satellite Imaging : Remote Sensing
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Ultra-sound Imaging of Brain
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Steady-State One-Dimensional Conduction Assume a homogeneous medium with invariant thermal conductivity ( k = constant) : For one-dimensional steady state conduction with no energy generation, the heat equation reduces to : One dimensional Transient conduction with heat generation.
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Steady-State One-Dimensional Conduction For one-dimensional heat conduction in a variable area geometry. We can devise a basic description of the process. The first law in control volume form (steady flow energy equation) with no shaft work and no mass flow reduces to the statement that Q = 0 for all surfaces. From Fourier law of conduction, the heat transfer rate in at the left (at x) is:
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Taylor’s Theory of Continuum For a function converging & well behaving… For a pure steady state conduction:
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Substitute Fourier’s law of conduction:
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If k is constant (i.e. if the material is homogeneous and properties of the medium are independent of temperature), this reduces to Pure radial conduction through A Sphere.
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Surface area of a sphere at r
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Heat transfer through a plane slab
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Isothermal Wall Surfaces
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Wall Surfaces with Convection Boundary conditions:
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Wall with isothermal Surface and Convection Wall Boundary conditions:
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Electrical Circuit Theory of Heat Transfer Thermal Resistance A resistance can be defined as the ratio of a driving potential to a corresponding transfer rate. Analogy: Electrical resistance is to conduction of electricity as thermal resistance is to conduction of heat. The analog of Q is current, and the analog of the temperature difference, T1 - T2, is voltage difference. From this perspective the slab is a pure resistance to heat transfer and we can define
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The composite Wall The concept of a thermal resistance circuit allows ready analysis of problems such as a composite slab (composite planar heat transfer surface). In the composite slab, the heat flux is constant with x. The resistances are in series and sum to R = R 1 + R 2. If T L is the temperature at the left, and T R is the temperature at the right, the heat transfer rate is given by
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Wall Surfaces with Convection Boundary conditions: R conv,1 R cond R conv,2 T1T1 T2T2
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R conv,1 R cond R conv,2 T1T1 T2T2
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