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Heat Conduction
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
In the previous section, standard coordinate systems including Cartesian and radial coordinates were presented. For more complex geometrical configurations, curvilinear coordinates may be used. Curvilinear coordinates refer to a coordinate system in which the coordinate lines may be curved. The coordinates can be obtained by a mapping from Cartesian coordinates by a coordinate transformation that is locally invertible (a one-to-one mapping from any point in the Cartesian coordinates to curvilinear coordinates). Common examples of curvilinear coordinate systems include cylindrical and spherical coordinates. These coordinates are illustrated in Figure 2.10. Other examples include ellipsoidal, toroidal, bispherical, and oblate spheroidal coordinates. Ellipsoidal coordinates are the most generalized coordinate system from which other curvilinear coordinate systems are special cases.
Evaporation of oblate spheroidal droplets: A theoretical analysis
Published in Chemical Engineering Communications, 2018
The heat equation (Equation (4)) could be used for any orthogonal coordinate system (such as a spherical or cylindrical system) once the metric coefficients are defined (Moon and Spencer, 1971). To construct an oblate coordinate system with respect to a rectangular coordinate system at (0, 0, 0), the positive z-axis is the axis line, the x, y plane is the equatorial plane and angles are measured from the positive x-axis in a clockwise direction with respect to z-axis. The metric coefficients for an oblate spheroid from Equation (3) are used to transform Equation (4) into oblate spheroidal coordinates. In a two-dimensional oblate orthogonal coordinate system, the coordinate ω is eliminated because of its rotational symmetry around the z-axis. Then, Equation (4) takes the following form:where α is the thermal diffusivity of the droplet,
Point set registration for reduced geometry mismatch during estimation of mass transfer properties in osmotic dehydration of complex-shaped foods
Published in Drying Technology, 2020
K. H. Estévez-Sánchez, J. E. González-Pérez, C. E. Ochoa-Velasco, M. A. García-Alvarado, D. Cruz-González, A. Sampieri, I. I. Ruiz-López
The solution of the diffusion equation for any simple separable orthogonal coordinate system, such as the oblate spheroidal coordinates, is always,[23] where should satisfy a given equation resulting from applying the corresponding boundary conditions. Thus, it is expected that an average solution will have the general form