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PS_5_2006S08

# PS_5_2006S08 - V2 1 2.006 PS 6 S08 MASSACHUSETTS INSTITUTE...

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Unformatted text preview: V2 1 2.006 PS 6 S08 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING PROBLEM SET 5 2.006 DUE: * Tuesday* , March 11, 2008 Problem 0 Read Chap 11 in notes, Incropera and Dewitt covers much of the same material in Chapters 7 and 8. Problem 1 In class, a model for the heat transfer to a fully developed laminar flow in a circular tube (radius R) was discussed. The fluid flowing through the tube was presumed to have a thermal conductivity k, a specific heat c p and a mass flow rate ˙ m . The heat flux q through the tube wall was assumed uniform. It was shown that the first law, if appropriately written, results in a partial differential equation for the temperature of the fluid, T(r,x) of the form: ¡ 1 r ¢ ¢ r r ¢ ¢ r £ ¤ ¥ ¦ § ¨ T = © ¢ T ¢ x where ¡ is the thermal diffusivity, x is the axial distance down the tube, r is the radial coordinate, T is the temperature and is a function of r and x, and ¡ is the local velocity. Given that the surface temperature of the pipe is T s (x). a) Please show explicitly (do and show the algebra/calculus) that the temperature distribution in the can be written as T = T s ( x ) ¡ 4 q kR 3 R 2 16 ¡ r 2 4 ¡ r 4 16 R 2 ¢ £ ¤ ¥ ¦ § . b) From this, using the definition of bulk temperature and the relation derived in class for the bulk temperature (the enthalpy-averaged temperature) as a function of x, please show that the temperature distribution can be written as T ( r , x ) = T b (0) + q 2 ¡ R ˙ m¡c p x + 11 24 qR k ¢ 4 q kR 3 R 2 16 ¢ r 2 4 + r 4 16 R...
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PS_5_2006S08 - V2 1 2.006 PS 6 S08 MASSACHUSETTS INSTITUTE...

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