Steady-State Conduction and Shape Factors

In previous sections of this chapter we discussed steady-state heat conduction in one direction. In many cases, however, steady-state heat conduction is occurring in two directions i.e., two-dimensional conduction is occurring. The two-dimensional solutions are more involved and in most cases analytical solutions are not available. One important approximate method to solve such problems is to use a numerical method discussed in detail in Section 4.15. Another important approximate method is the graphical method, which is a simple method that can provide reasonably accurate answers for the heat-transfer rate. This method is particularly applicable to systems having Isothermal boundaries. 

Draw a model to scale of the two-dimensional solid. Lable the isothermal boundaries. In Fig. 4.4-1, Ti and T2 are isothermal boundaries. 

Select a number N that is the number of equal temperature subdivisions between the isothermal boundaries. In Fig. 4.4-1, N = 4 subdivisions between T, and Tj. Sketch in the isotherm lines and the heat flow or-flux lines so that they are perpendicular to each other at the intersections. Note that isotherms are perpendicular to adiabatic (insulated) boundaries and also lines of symmetry. 

Keep adjusting the isotherm and flux lines until for each curvilinear square the condition Ax = Ay is satisfied. 

In order to calculate the heat flux using the results of the graphical plot, we first assume unit depth of the material The heat flow q through the curvilinear section shown in Fig. 4.4-1 is given by Fourier s law. 

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