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Bounds for the Thermal Conductance of Body of Rotation

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A mathematical heat transfer model is developed for the steady-state heat transfer through a homogeneous isotropic body of rotation. The developed model is used to obtain estimations for two-sided thermal heat transfer conductance in case of a special type of body of rotation. The body of rotation considered is bounded by the coordinate surfaces of an orthogonal curvilinear coordinate system. The equations of the Fourier’s theory of heat conduction in solid body are used to formulate the corresponding thermal boundary value problem. The studied steady-state heat conduction problem is axisymmetric. The determination of the heat flow is based on the concept of the thermal conductance the inverse of which is the thermal resistance. The exact (strict) value of thermal conductance is known only for bodies with very simple shapes, therefore, the principles and the methods that can be used for creating lower and upper bounds to the numerical value of thermal conductance are important. The derivation of the upper and the lower bound formulae for the heat conductance of axisymmetric ring-like body is based on the two types of Cauchy–Schwarz inequality. The condition of equality of the derived lower and upper bounds is examined. Several examples illustrate the applications of the derived upper and lower bound formulae. The presented method can be used not only for the estimation of thermal conductance. For example, similar formulation can be given to obtain two-sided bounds for the electrical resistance of solid body conductor in the case of spatially steady flow of electric charges.
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Heat Conduction; Steady-State; Body of Rotation; Thermal Conductance; Bounds

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