Conduction heat transfer shape factors

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... 

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. 

The number of experimental factors which influence the results increases considerably when thermogravimetry is combined with other techniques such as DTA, gas chromatography46, mass spectrometry, X-ray etc. A systematic discussion of all these additional factors would lead too far, therefore only a representative example will be discussed here. One of the often-applied multiple techniques is the combination TG-DTA. Besides the actual thermal reactions of the sample, the important factors in DTA are the heat capacity and the thermal conductivity of the sample. Optimum heat transfer is required for such thermoanalytical measurements therefore the shape of the sample and its contact with the crucible is of special importance. 

In a series of papers, Derby and Brown (144, 149-152) developed a detailed TCM that included the calculation of the temperature field in the melt, crystal, and crucible the location of the melt-crystal and melt-ambient surfaces and the crystal shape. The analysis is based on a finite-ele-ment-Newton method, which has been described in detail (152). The heat-transfer model included conduction in each of the phases and an idealized model for radiation from the crystal, melt, and crucible surfaces without a systematic calculation of view factors and difiuse-gray radiative exchange (153). 

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

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). 

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. 

Hohne (145) pointed out that the function principle of DSC can give rise to calibration errors in case of phase transitions disturbing the steady-state conditions. The cause of this problem is the temperature dependence of the coefficients of heat transfer, leading to weak nonlinearity of the calorimeter. This results in a dependence of the calibration factor on parameters such as mass and thermal conductivity of the sample, heating rate, peak shape, and temperature. By theoretical considerations and calculations, the uncertainty of the calibration factor due to the variation of sample parameters can be 1-5%, depending on the temperature and the instrument involved. 

In many problems concerning heat transfer, the thermal conductivity X of the materials is an important factor. The rate at which heat is transmitted through glass by conduction depends on size and shape, on the difference in temperature between the two faces and on the composition of the material. Thermal conductivity is commonly ex-... 

Powder particle size, shape, and size-distribution are important factors in determining the moldability of a material. Heat is transferred to the powder by conduction with other particles and the mold and by convection with the surrounding air. 

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