One-Dimensional, Time-Dependent

A mountain range can be represented as a periodic topog raphy with a wavelength of 100 km and an amplitude of 1.2 km. Heat flow in a valley is measured to be 46 mW m−2. If the atmospheric gradient is 6.5 K km−1 and k = 2.5 Wm−1 K−1, determine what the heat flow would have been without topography; that is, make a topographic correction.

Heat Conduction Many of the important geological problems involving heat conduction are time dependent. Examples that we consider later are the cooling of intrusive igneous bodies, the cooling of the oceanic lithosphere, erosion or sedimentation effects on temperature, and others. Volumetric heat production usually plays a minor role in these phenomena, and we accordingly assume H = 0.

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In addition, it is adequate to consider heat conduction in one direction only. If there are not heat sources in the medium, a net heat flow out of the slab illustrated in Figure 4–5 must reduce its temperature. The specific heat c of the medium is the energy required to raise the temperature of a unit mass of material by one degree. Thus, an element of the slab of thickness δy and unit cross-sectional area requires an energy flow per unit time given by ρc ∂T ∂t δy 270 Heat Transfer to maintain a temperature change at the rate ∂T/∂t(ρδy is the slab mass per unit cross-sectional area and ρcδy is the slab’s heat capacity per unit cross-sectional area). Thus we can equate the right side of Equation (4–10) with −δy ρc∂T/∂t, since a net heat flow out of the slab leads to a decrease in slab temperature ρc ∂T ∂t = k ∂2T ∂y2 . (4.67) Equation (4–67) is the basic equation governing the time and spatial vari- ations of the temperature when heat is transferred in one dimension by conduction. Partial derivatives are required because T is a function of both time and space. We can rewrite Equation (4–67) in the form ∂T ∂t = κ ∂2T ∂y2 , (4.68) where κ, the thermal diffusivity, is κ = k ρc . 

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