Boundary conditions heal flux

The heat conduction equation is first order in time, and thus the initial condition cannot involve any derivatives (it is limited to a specified temperature). However, the heal conduction equation is second order in space coordinates, and thus a boundary condition may involve first derivalives at the boundaries as well as specified values of temperature. Boundary conditions most commonly encountered in practice are the specified temperature, specified heat flux, convection, and radiation boundary conditions. 

Some surfaces are commonly insulated in practice in order to minimize heat loss (or heal gain) through them. Insulation reduces heat transfer but does not loialf eliipinate it unless its thickness is infinity. However, heat transfer through a properly insulated surface can be taken to be zero since adequate insulation reduces heat transfer through a surface to negligible levels. Therefore, a well-insulated surface can be modeled as a surface with a specified heat flux of zero, llien the boundary condition on a perfectly insulated surface (at X - 0, for example) can be expressed as (Fig. 2-30)... 

W In this case different heal fluxes are specified at the two boundaries. The application of the boundary conditions gives... 

Consider a spherical container of inner radius r, outer radius rj, and thermal conductivity k. Express the boundary condition on the inner surface of the container for steady one-dimensional conduction for the following cases (a) specified temperature of 50°C, (b) specified heat flux of 30 W/m toward the center, (c) convection to a medium at 7. with a heal transfer coefficient of/i. 

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