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Heat Conduction Handout

Heat Conduction Handout - Department of Chemical...

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Department of Chemical Engineering & Materials Science University of California Davis Chemical Engineering Laboratory – ECH 155A Heat Conduction in Solid Rods Experiment Goals 1. Determine the transient and steady state temperature profiles in solid rods heated in one end by a constant temperature energy source. 2. Compare experimental results with theoretically predicted temperature profiles. Theoretical Background A long, solid rod of uniform material and circular cross section, which is initially at a uniform temperature equal to that of the ambient air, is suddenly heated at one end by a constant temperature energy source. It is assumed that: 1. All physical properties of the rods (e.g., thermal conductivity, density, heat capacity, etc.) are independent of temperature. 2. The rods are long. 3. Heat is only conducted along the length of the rod; the temperature is uniform across a radial cross-section. 4. The film heat transfer coefficient is independent of axial position. The conservation of energy then becomes T t = α 2 T x 2 - β (T - T a ) (1) Here, T denotes the temperature, which is a function of time, t, and the distance along the rod, x. The thermal diffusivity is α and β is related to the heat transfer coefficient (see Nomenclature). The assumptions listed above can be transformed into the following initial and boundary conditions: 1. Initially uniform temperature: t 0 T=T a for x 0 2. Constant temperature energy source: x=0 T=T s for t>0 3. Long Rod: x= T=T a for t>0 Equation (1) can be solved subject to these boundary conditions to obtain the transient temperature profiles 1 2 2 2 x x a s a T T x x e erfc t e erfc t T T t t β β α α β β α α - - = + + - - (2) At long times, as t , the steady state profiles can be obtained from Eq. (2).
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