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Mathematical Models for Modes of Heat Action P M V Subbarao Professor Mechanical Engineering Department Tools to select Means of Heat Interactions between System and Surroundings ….

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The Pentium 4 Processor

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Heat Sinks for Pentium 4

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Pentium 4 While Performing

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Statement of Fouriers Law The (mod of) heat flux, q, (the flow of heat per unit area and per unit time), at a point in a medium is directly proportional to the temperature gradient at the point. x T Temperature gradient across the slab of thickness x: The heat flux across the slab

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Local Heat flux in a slab: Global heat transfer rate:

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Mathematical Description Temperature is a scalar quantity. Heat flux is defined with direction and Magnitude : A Vector. Mathematically it is possible to have: Using the principles of vector calculus:

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Further Physical Description Will k be same in all directions? Why k cannot be different each direction? Why k cannot be a vector? Will mathematics approve this ? What is the most general acceptable behavior of k, approved by both physics and mathematics?

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Anisotropic Materials x,y z

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Thermal Image of Laptop Casing

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Graphite Covering

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Thermal Image of Laptop Casing with Graphite cover

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Most General form of Fourier Law of Conduction We are at cross roads !!!!!

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Physically & mathematically Feasible Model Taking both physics and mathematics into consideration, the most feasible model for Fouriers Law of conduction is: Thermal conductivity of a general material is a tensor.

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Surprising Inventions !!!

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Radiative Mode of Heat Transfer Any body (> absolute zero) emits radiation at various wavelengths. Transparent bodies radiate energy in spherical space. Non-transparent bodies radiate energy in hemi-spherical space. The radiation energy emitted by a body is distributed in space at various wavelengths. This complex phenomenon requires simplified laws for engineering use of radiation.

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Planck Radiation Law The primary law governing blackbody radiation is the Planck Radiation Law. This law governs the intensity of radiation emitted by unit surface area of a blackbody as a function of wavelength for a fixed temperature. h = 6.625 X 10 -27 erg-sec (Planck Constant) K = 1.38 X 10 -16 erg/K (Boltzmann Constant) C = Speed of light in vacuum The Planck Law can be expressed through the following equation.

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Stefan-Boltzmann Law The maximum emissive power at a given temperature is the black body emissive power (E b ). Integrating this over all wavelengths gives E b. Driving forces: Heat transfer by radiation is driven by differences in emissive power (proportional to T 4.

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The total energy emitted by a real system, regardless of the wavelengths, is given by: where ε sys is the emissivity of the system, A sys-surface is the surface area, T sys is the temperature, and σ is the Stefan-Boltzmann constant, equal to 5.67×10 -8 W/m 2 K 4. Emissivity is a material property, ranging from 0 to 1, which measures how much energy a surface can emit with respect to an ideal emitter (ε = 1) at the same temperature Radiation from a Thermodynamic System

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Radiative Heat Transfer between System and Surroundings Consider the heat transfer between system surface with surroundings, as shown in Figure. What is the rate of heat transfer into system surface ? This radiation is emitted in all directions, and only a fraction of it will actually strike system surface. This fraction is called the shape factor, F. To find this, we will first look at the emission from surroundings to system. Surrounding Surface emits radiation as described in

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The amount of radiation striking system surface is therefore: The only portion of the incident radiation contributing to heating the system surface is the absorbed portion, given by the absorptivity α B : Above equation is the amount of radiation gained by System from Surroundings. To find the net heat transfer rate for system, we must now subtract the amount of radiation emitted by system:

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The net radiative heat transfer (gain) rate at system surface is Similarly, the net radiative heat transfer (loss) rate at surroundings surface is What is the relation between Q sys and Q sur ?

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