A survey of heat conduction problems

Abstract

Several forms of thermal conductivity apparatus are described which have been devised for the study of diverse materials ranging from a sheet of mica to a wall section weighing half a ton. The materials are broadly classified into four groups: (1) Materials of very low thermal conductivity, such as cold storage insulators. (2) Materials supplied in the form of thin sheets and those employed in building construction. (3) Refractories and materials employed in furnace construction. (4) Pure metals and alloys. Samples of baked slab cork are shown with a conductivity as low as 0.00007 c.g.s. unit. The insulating value of such materials is due to the fact that they subdivide the air space into a large number of minute air cells. The theory of heat conduction through granular material is an interesting problem. An attempt to work it out mathematically has been made by M. Smoluchowski dagger. The wide divergence between the results obtained by experiment and those calculated from theoretical considerations indicate that the mechanism of the heat transfer from solid to gas is one which requires further study. In the testing of materials of low thermal conductivity by the hot plate method attention has to be given to heat leakage from edges and corners. The simplest procedure is to eliminate the by the use of a guard plate, but it is also possible to calculate an approximate correction for the edge effect when using the simple hot plate method. A mathematical problem awaiting solution with numerous practical applications is the adaptation of the method of conformal representation to three-dimensional problems. Sixty years have now elapsed since the publication of the important papers by Schwarz and Christoffel, and in this interval numerous papers have appeared demonstrating what a powerful tool conformal representation is for the solution of two-dimensional problems. Substances in the form of thin discs can be tested by the divided bar method, the correction for the thermal resistivity of the two mercury films being obtained by independent experiments using a thin disc of iron instead of the specimen. This apparatus is not well adapted for the study of poor heat-conductors, owing to the fact that the temperature drop across the specimen is then large and the gradient along the two bars small. The electrically heated plate, sandwiched between two discs of the material with water-cooled plates on either side, is an apparatus which can be adapted to various substances obtainable in sheet form. A form of this apparatus devised for the study of the of pressure on the flow of heat through a pile of paper-covered iron stampings is illustrated. In this test the thermal resistance across the laminations is measured. For the determination of the heat flow along the laminated material a bar about 15 in. long of square section, 2 in. by 2 in., is built up. One end is electrically heated, while the other dips into a pot of mercury carrying a cooling coil. From the temperature gradient and the heat flow the conductivity can be calculated. The apparatus for the study of refractories consists of a furnace made up of silit rod heaters which can raise one face of the slab under test to any temperature up to 1000° C. On the top of the slab is a water-flow calorimeter fitted with a guard ring. The calorimeter and guard ring are made of thick slab brass with grooves milled in the back surfaces for water circulation. To reduce heat transfer between calorimeter and guard ring the edges of the plates are chamfered off so that the edges on the brick are near together, whilst the rear parts are well separated. For the study of metals and alloys the guard tube method for the elimination of heat loss sideways is to be recommended in preference to Forbes' procedure. The space between guard tube and specimen is packed with a powder of low thermal conductivity. Both specimen and guard tube can be attached to a cylindrical block containing a heating coil. The heat flow is determined with a cooling coil fixed to the other end of the specimen. Water flows round the end of the specimen and the guard ring so that the same gradient is maintained automatically in specimen and guard tube. In some experiments the guard tube is dispensed with and the heat leakage through the lagging material calculated approximately. In the study of two groups of alloys - aluminium alloys and bronzes - it has been found that the two groups give values of the Lorentz coefficient approximating to the values for the pure metals which form the principal constituents of the alloys - namely, 5.5 for the aluminium alloys and 5.9 for the copper alloys.