Conduction of Heat in Solids

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Carslaw and Jaeger. 1947. Conduction of Heat in Solids. Oxford University Press, New York. 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

CARSLAW, H.S., J.C.JAEGER Conduction of heat in solids. Oxford Claredon Press 1959. 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

New York (1933) 7) R.R. Wenner Thermochemical Calculations , McGraw Hill, New York (1941) 8) J.M. Cork Heat , J. Wiley, New York (1942) 9) J. Reilly W.N. Rae Physicochemical Methods , Van Nostrand, New York (1943) 10) H.S. Carslaw J.C. Jaeger Conduction of Heat in Solids , Clarendon Press, Oxford (1947) 11) M. Jacob, Heat Transfer , J. Wiley Sons, New York, V. 1 (1949) 12) D.Q. Kern Process Heat Transfer , McGraw Hill, New York (1951) 13) F. Reif... 

In the theory of diffusion-controlled reactions it is called the radiative boundary condition . This quaint name originated from H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford 1947) p. 13. 

Car slaw AS, Jaeger JC (1959) The conduction of heat in solids, 2nd edn. Oxford Clarendon Press, Oxford... 

H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids", 2nd Ed. Oxford University Press, (Oxford, 1959)... 

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