Mathematics, 03.09.2020 04:01, garcikyl000
Consider a thin one-dimensional rod without sources of thermal energy whose lateral surface area is not insulated. (a) Assume that the heat energy flowing out of the lateral sides per unit surface area per unit time is w(x, t). Derive the partial differential equation for the temperature u(x, t). (b) Assume that w(x, t) is proportional to the temperature difference between the rod u(x, t) and a known outside temperature Ξ³(x, t).Ξ³(x, t). Derive crho βu/βt = β/βx(K0βu/βx)βPA[u(x, t)βΞ³(x, t)]h(x)...(1.2.15)crho [u(x, t)βΞ³(x, t)]h(x)...(1.2.15) where h(x) is a positive x-dependent proportionality, P is the lateral perimeter, and A is the cross-sectional area. (c) Compare (1.2.15) with the equation for a one-dimensional rod whose lateral surfaces are insulated, but with heat sources. (d) Specialize (1.2.15) to a rod of circular cross section with constant thermal prop erties and 0β outside temperature. (e) Consider the assumptions in part (d). Suppose that the temperature in the rod is uniform [i. e., u(x, t) = u(t)]. Determine u(t) if initially u(0)=u0.
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Mathematics, 21.06.2019 16:30, happy121906
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Consider a thin one-dimensional rod without sources of thermal energy whose lateral surface area is...
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