Slabs heat conduction into

Heisler charts Aset of graphical plots used to determine the time taken for thermal penetration by heat conduction into a soUd body by heating or cooling at its surface. The plots are prepared for standard geometric shapes such as slabs, cylinders, and spheres with the Fourier number on the x-axis and a dimensionless temperature on the y-axis. The lines represent the reciprocal of the Biot number. They are named after M. P. Heisler, who com -piled them in 1947. 

Fig. 20.1-2. Unsteady heat conduction into a semi-infinite slab. The temperature profile in this case is an error function, just like the concentration profile in Section 2.3. This profile depends on the variable z/v/4at, where a (= kr/pCp) is the thermal diffusivity.

Numerical calculation methods for unsteady-state heat conduction are similar to numerical methods for steady state discussed in Section 4.15. The solid is subdivided into sections or slabs of equal length and a fictitious node is placed at the center of each section. Then a heat balance is made for each node. This method differs from the steady-state method in that we have heat accumulation in a node for unsteady-state conduction. 

To calculate heat fluxes or temperature profiles, we make energy balances and then combine these with Fourier s law. The ways in which this is done are best seen in terms of two examples heat conduction across a thin film and into a semi-infinite slab. The choice of these two examples is not casual. As for diffusion, they bracket most of the other problems, and so provide limits for conduction. 

The limits of heat conduction across a thin film and into a thick slab are the two most important cases of a rich variety of examples. This variety largely consists of solutions of Eq. 20.1-16 for different geometries and boundary conditions. The geometries include slabs, spheres, and cylinders, as well as more exotic shapes like cones. The boundary conditions are diverse. For example, they include boundary temperatures that vary periodically because this is important for diurnal temperature variations of the earth. They include boundary conditions in which the heat flux at the surface is related to the temperature of the surroundings, Tsurrl for example. 

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. 

It is this method which is currently being used to manufacture sapphire boules as large as 18 inches in diameter. The boule is then cut into slabs which are polished and used for UV transmitting windows, substrates for various electronic devices and even non-scratching faces on your watch. Note that sapphire is about as hard as diamond and has a heat-transmitting capability almost equal to many metals, while remaining essentially chemically inert and electrically non-conductive. 

Unsteady-state conduction in a cylinder. In deriving the numerical equations for unsteady-state conduction in a flat slab, the cross-sectional area was constant throughout. In a cylinder it changes radially. To derive the equation for a cylinder. Fig. 5.4-3 is used where the cylinder is divided into concentric hollow cylinders whose walls are Ax m thick. Assuming a cylinder 1 m long and making a heat balance on the slab at point n, the rate of heat in — rate of heat out = rate of heat accumulation. 

Although in refractory practice there are hundreds of heat insulation materials, the list of heat insulation materials for the lining of reduction cells is rather limited. For one thing, economic considerations add some limitations, but for another, the heat insulation materials in reduction cells should withstand mechanical compression loads without deformation at temperatures up to 900 °C for a long time, and numerous inexpensive fiber heat insulation materials don t correspond to this requirement. In the Hall-Heroult reduction cell, the heat insulation materials should withstand the pressure of the layer of the electrolyte, the layer of molten aluminium, cathode carbon blocks (taking into account collector bars), and the refractory layer. Currently, only four or five heat insulation materials are used in the lining of reduction cells diatomaceous (moler) and perlite bricks, vermiculite and calcium silicate blocks (slabs), and sometimes lightweight fireclay bricks (but their thermal conductivity is relatively big, while the cost is not small) and fiber fireclay bricks. 

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