Heat conduction equation rectangular coordinates

Application of the law of conservation of energy to a three-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equation for unsteady-state conduction in rectangular coordinates. 

Eq. 2-38 is the general heat conduction equation in rectangular coordinates. In the case of constant thermal conductivity, it reduces to... 

The general heat conduction equations in spherical coordinates can be obtained from an energy balance on a volume element in spherical coordinates, shown in Fig, 2-24, by following the steps outlined above. It can al.so be obtained directly from Eq. 2-38 by coordinate transformation using the following relations between the coordinates of a point in rectangular and spherical coordinate systems ... 

To use Fourier s law of heat conduction, a thermal balance must first be constructed. The energy balance is performed over a thin element of the material, x to x + Ax in a rectangular coordinate system. The energy balance is shown in equation 13 ... 

One-Dimensional Conduction Lumped and Distributed Analysis The one-dimensional transient conduction equations in rectangular (b = 1), cylindrical (b = 2), and spherical (b = 3) coordinates, with constant k, initial uniform temperature 7), S = 0, and convection at the surface with heat-transfer coefficient h and fluid temperature 77, are... 

One-Dimensional Conduction Many heat-conduction problems may be formulated into a one-dimensional or pseudo-one-dimensional form in which only one space variable is involved. Forms of the conduction equation for rectangular, cylindrical, and spherical coordinates are, respectively. 

We start this chapter with a description of steady, unsteady, and multidimensional heat conduction. Then we derive the differential equation that governs heat conduction in a large plane wall, a long cylinder, and a sphere, and generalize the results to three-dimensional cases in rectangular, cylindrical, and spherical coordinates. Following a discussion of the boundary conditions, we present tlie formulation of heat conduction problems and their solutions. Finally, we consider lieat conduction problems with variable thermal conductivity. 

In the last section we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Most heat transfer problems encountered iu practice can be approximated as being onedimensional, and we mostly deal with such problems in tliis text. However, this is not always the case, and sometimes we need to consider heat transfer in other directions as well. In such cases heal conduction is said to be multidimensional, and in this section we develop the governing differential equation in such systems in rectangular, cylindrical, and spherical coordinate systems. 

S- 105 Starting with an energy balance on Ihe volume ele-inent, obtain the steady three-dimensional finite difference equation for a general interior node in rectangular coordinates for 7 x, y, z) for the case of constant thermal conductivity and uniform heat generation. 

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