Conduction equations: in cylindrical coordinates

The theoretical model assumes a line heat source dissipating heat radially into an infinite solid, initially at uniform temperature. The fundamental heat conduction equation in cylindrical coordinates, assuming uniform radial heat transfer, is [3] ... 

Beginning with the three-dimensional heat-conduction equation in cartesian coordinates [Eq. (l-3a)], obtain the general heat-conduction equation in cylindrical coordinates [Eq. (1-36)]. 

The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates, shown in Fig. 2-23, by following the steps just outlined. It can also be obtained directly from Eq. 2-38 by coordinate transformation u ing the... 

Starting with an energy balance on a ring-shaped volume element, derive the two-dimensional steady heat conduction equation in cylindrical coordinates for T(r, z) for the caf.se of constant thermal conductivity and no heat generation./... 

The heat conduction equation in cylindrical coordinates is simplified to... 

The heal conduction equation in cylindrical or spherical coordinates can be nondimensiooalized in a similar way. Note that nondimeosionalization... 

Densification behaviors of electrical conducting Cu and insulator AI2O3 with SPS have been compared and modeled, with the portion of the system schematically shown Fig. 6.24 [39]. In this modeling study, stainless steel electrodes (rams) are included in the model. The boundary conditions of axial cooling and the consequent temperature distribution change dramatically. The mathematical equations used in the modeling, i.e., the enthalpy and current density conservation equations in cylindrical coordinates applied to the elements, are solved in full 3D version through CFD-Ace (FEM). Temperature dependences of thermophysical properties for aU materials have been considered. 

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. 

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... 

The steady state heat conduction equation with heat source in cylindrical coordinate system is given by... 

Eacc is the accumulation term, the convection term, E ond the conduction term, and Ediss the dissipation term. Equation 5.5(c) is an expression of Eourier s law of heat transfer see also Section 5.3.1. In cylindrical coordinates, only the convection, conduction, and dissipation terms change ... 

On the other hand, the energy balance due to convection and conduction results from an energy balance in cylindrical coordinates. In the limit and using Fourier equation for the heat flux, q, we obtain Equation 4.36 ... 

For the plug flow region and the refractory wall, the discretized equations are the same as for the active region of the bed except that they are in cylindrical coordinates. For the radial direction, only the conduction component is considered and the coefficient of the algebraic (discretized) equation is equal to, for example,... 

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) ... 

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. 

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. 

In the two most important coordinate systems, cartesian coordinates x, y, z, and cylindrical coordinates r, (p, z the heat conduction equation takes the form... 

Its cylindrical nature means that cylindrical coordinates are the most convenient for analysing heat conduction in the fuel pin. A formal statement of equation (A8.4)... 

A circular waveguide with inner radius a is shown in Fig. 4.14. Here the axis of the waveguide is aligned with the z axis of a circular-cylindrical coordinate system, where p and are the radial and azimuthal coordinates, respectively. If the walls are perfectly conducting and the dielectric material is lossless, the equations for the TE , modes are... 

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. 

The first equation is listed in rectangular, cylindrical, and spherical coordinates in Table 2.5. For incompressible Newtonian fluids with constant thermal conductivity, Eq. 2.9-16 reduces to ... 

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). 

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. 

Equation (2) represents the heat flow into the volume V and can be derived from Eq. (11) in Fig. 1.2. The symbols have the standard meanings p is the density and Cp, the specific heat capacity. Standard techniques of vector analysis now dlow the heat flow into the volume V to be equated with the heat flow across its surface. This operation leads to the Foimer differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity, which is equal to the thermal conductivity k divided by the density and specific heat capacity. Its dimension is m /s. The Laplacian operator, v2, is 32/3j2 + 2/ 2 + a2/aj2 where x,y and z are the space coordinates. In the present example of cylindrical symmetry, the Laplacian operator, operating on temperature T, can be represented as — i.e., the... 

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