Heat conduction shape factor

In the heat transfer literature the corresponding quantity, hoAjK, is sometimes called the conduction shape factor. 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

Conduction shape factors have been determined for a number of configurations encountered in practice and are given in Table 3-7 for some common cases. More comprehensive tables are available in the literature. Once the value of the shape factor is known for a specific geometry, the total steady heat transfer rate can be determined from the equation above using the specified two constant temperatures of the two surfaces and the thermal conductivity of the medium between them. Note that conduction shape factors are applicable only when heat transfer between the two surfaces is by conduction. Therefore, they cannot be used when the medium between the surfaces is a liquid or gas, which involves natural or forced convection currents. 

Conduction shape factors Sfor several configurations for use in Q = frS(T, - T2) to determine the steady rate of heat transfer through a medium of thermal conductivity k between the surfaces at temperatures Ti and... 

Shape Factor and Thermal Resistance in Orthogonal Curvilinear Coordinates. The definition of thermal resistance of a system (total temperature drop across the system divided by the total heat flow rate) yields the following general expression for the thermal resistance R and the conduction shape factor S ... 

A. V. Hassani and K. G. T. Hollands, Conduction Shape Factor for a Region of Uniform Thickness Surrounding a Three-Dimensional Body of Arbitrary Shape, Journal of Heat Transfer, Vol. 112, pp. 492-495,1990. 

Conduction shape factor Dimensionless factor used to account for the geometrical effects in steady-state heat conduction between surfaces at different temperatures. 

Shrivastava, D. and R.B. Roemer, An analytical study of Poisson conduction shape factors for two thermally significant vessels in a finite, heated tissue. Physics in Medicine and Biology, 2005. 50(15) 3627-3641. 

The second model, proposed by Frank-Kamenetskii [162], applies to cases of solids and unstirred liquids. This model is often used for liquids in storage. Here, it is assumed that heat is lost by conduction through the material to tire walls (at ambient temperature) where the heat loss is infinite compared to the rate of heat conduction through the material. The thermal conductivity of the material is an important factor for calculations using this model. Shape is also important in this model and different factors are used for slabs, spheres, and cylinders. Case B in Figure 3.20 indicates a typical temperature distribution by the Frank-Kamenetskii model, showing a temperature maximum in the center of the material. 

Once the electric potential is impressed on the paper, an ordinary voltmeter may be used to plot lines of constant electric potential. With these constant-potential lines available, the flux lines may be easily constructed since they are orthogonal to the potential lines. These equipotential and flux lines have precisely the same arrangement as the isotherms and heat-flux lines in the corresponding heat-conduction problem. The shape factor is calculated immediately using the method which was applied to the curvilinear squares. 

Andrews, R. V. Solving Conductive Heat Transfer Problems with Electrical-analogue Shape Factors, Chem. Eng. Prog., vol. 51, no. 2, p. 67, 1955. 

Sunderland, J. E., and K. R. Johnson Shape Factors for Heat Conduction through Bodies with Isothermal or Convective Boundary Conditions, Trans. ASHAE, vol. 70, pp. 237-241, 1964. 

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). 

Note. V is the molar volume, JVyi is Advogadro s number, is the equilibrium concentration, D is the diffusion coefficient, sub-s surface, hHfiB the heat of fusion, t) is the Damkohler number. Ah is the thermal conductivity, i die area shape factor for surface nuclei , y, is the distance between steps, n is the equilibrium surface concentration, p = 1 - o-JS is one minus the maximum surface supersaturation divided by the solution supersaturation, and p is the density. ihG - pl- fPMpAkBT In S)... 

Many conduction problems involving one-, two-, and three-dimensional steady-state heat flows employ a shape factor S, defined by Equation (6.18) ... 

FIGURE 6.11 Shape factors for various steady-state conduction geometries. (From Parker, J. D., Boggs, S. H., and Blick, E. R, Introduction to Fluid Mechanics and Heat Transfer, Addison-Wesley, Reading, MA, 1969.)... 

G. K. Lewis, Shape Factors in Conduction Heat Flow for Circular Bars and Slabs with Internal Geometries, Int. J. Heat Mass Transfer, Vol. 11, pp. 985-992,1968. 

M. L. Ramachandra Murthy and A. Ramachandran, Shape Factors in Conduction Heat Transfer, British Chemical Engineering Design (12/5) 730-731,1967. 

Dendrites form if solidification is limited by diffusion. The dissipation of latent heat can be the rate-determining factor. As heat conduction is better in the liquid than in the solid phase, there is spatial anisotropy in the solidification rate and the solid assumes a ramified shape. Latent heat is more easily disposed of at the end of the dendrites than in depressions on the growing solid surface, and the tops of dendrites grow faster than the depressions in the surface. This positive feedback yields the same type of morphology as diffusion-limited aggregation, which is... 

This shape factor S has units of m and is used in two-dimensional heat conduction where only two temperatures are involved. The shape factors for a number of geometries have been obtained and some are given in Table 4.4-1. 

In Section 4.4 we discussed methods for solving two-dimensional heat-conduction problems using grap]iical procedures and shape factors. In this section we consider analytical and numerical methods. 

---