Heat conduction equation spherical coordinates

The governing equation for heat conduction in the cross section of a rotating wall can be approximated by that of transient heat conduction in spherical coordinates as (Gorog et al., 1982)... 

Cylindrically symmetric and spherically symmetric heat conduction problems. In explorations of many physical processes such as diffusion or heat conduction it may happen that the shape of available bodies is cylindrical. In this view, it seems reasonable to introduce a cylindrical system of coordinates (r, ip, z) and write down the heat conduction equation with respect to these variables (here x = r) ... 

Write the simplified heat-conduction equation for (a) steady one-dimensional heat flow in cylindrical coordinates in the azimuth (< ) direction and (b) steady onedimensional heat flow in spherical coordinates in the azimuth (0) direction. 

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 ... 

The problem can be solved effectively by converting the convection-diffusion equation into the well studied heat conduction equation by introducing the stream fimction P as a new variable. In terms of the stream function the velocity components in spherical coordinates z and 0 are,... 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

Similar equations apply to cylindrical and spherical coordinate systems. Finite difference, finite volume, or finite element methods are generally necessary to solve (5-15). Useful introductions to these numerical techniques are given in the General References and Sec. 3. Simple forms of (5-15) (constant k, uniform S) can be solved analytically. See Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966, p. 180, and Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. For problems involving heat flow between two surfaces, each isothermal, with all other surfaces being adiabatic, the shape factor approach is useful (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 164). 

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. 

The initial conditions are at t = 0, T = To, andp = 0. The parameter n characterizes the dimensions of the volume for a parallel plate reactor n = 0 for a cylindrical reactor n = 1 and for a spherical reactor n = 2. In these equations, x is a space coordinate A. is the coefficient of thermal conductivity r is the characteristic size of the reactor k is the heat transfer coefficient and To is the initial temperature of the initial medium. 

This equation may be transformed into other systems of orthogonal coordinates, the most useful being the spherical polar system. (Carslaw and Jaeger, Conduction of Heat in Solids, gives details of the transformation.) When the operation is performed ... 

---