Consider an extended surface of rectangular cross section with heat flow in the longitudinal direction.

In this problem we seek to determine conditions for which the transverse (y-direction) temperature difference within the extended surface is negligible compared to the temperature difference between the surface and the environment, such that the one-dimensional analysis of Section 3.6.1 is valid.

(a) Assume that the transverse temperature distribution is parabolic and of the form where Ts(x) is the surface temperature and To(x) is the centerline temperature at any x-location. Using Fourier's law, write an expression for the conduction heat flux at the surface, qx(t), in terms of Ts and T0'

(b) Write an expression for the convection heat flux at the surface for the x-location. Equating the two expressions for the heat flux by conduction and convection, identify the parameter that determines the ratio (To - Ts) / (Ts - T∞).

(c) From the foregoing analysis, develop a criterion for establishing the validity of the one-dimensional assumption used to model an extended surface.

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