PS2_sol_S08 - Problem Set 2 (2.006, Spring 08) Problem 1...

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Unformatted text preview: Problem Set 2 (2.006, Spring 08) Problem 1 (a) There are 8 important parameters. There are 4 different dimensions appearing in the table. Try j = 4 and , , D , and k f as repeating parameters. ( ) ( ) ( ) ( ) 3 1 1 3 1 a b d c a b c d f D k ML ML T L MLT M L T = = : : 3 : 3 : M a b d L a b c d T b d d + + = + + = = = a b c d = = = = Now non-dimensionalize h , U , c p , and k s with the repeating variables. h: ( ) ( ) ( ) ( ) 3 1 3 1 1 3 1 a b d c a b c d f h D k M T M L M L T L M L T M L T = = : 1 : 3 : 3 3 : 1 M a b d L a b c d T b d d + + + = + + = = = 0, 1, 1 a b c d = = = = 1 f hD k = U: ( ) ( ) ( ) ( ) 1 3 1 1 3 1 a b d c a b c d f U D k L T M L M L T L M L T M L T = = : : 1 3 : 1 3 : M a b d L a b c d T b d d + + = + + = = = 1, 1, a c b d = = = = 2 UD = c p : ( ) ( ) ( ) ( ) 2 2 1 3 1 1 3 1 a b d c a b c d p f c D k L T M L M L T L M L T M L T = = : : 2 3 : 2 3 : 1 M a b d L a b c d T b d d + + = + + = = = 0, 1, 1 a c b d = = = = 3 p f c k = k s must have the dimension of k f . ' 4 s f k k = Parameter h U D f k p c s k Dimension 3 1 MT 1 LT 3 ML 1 1 ML T L 3 1 MLT 2 2 1 L T 3 1 MLT (b) 1 Nu f hD k = = , 2 Re UD = = , 3 Pr p f f p c k k c = = = = 1 ' 4 4 Bi s f f s hD k hD k k k = = = = Nu and Bi look similar but they have different thermal conductivity. ( ) Nu Re, Pr, Bi f = (c) 1 2 3 4 5 10000 20000 30000 40000 h (W/m 2 K) v (m/s) Water Therminol 60 Air It seems that each case follows a power function with different coefficient and exponent. (d) Air (Pr=0.700) Therminol 60 (Pr=81.1) Water (Pr=5.93) Re Nu Re Nu Re Nu 63.3 4.1 15.6 10.9 574.4 26.1 316.4 9.0 78.0 24.0 2872.0 58.1 632.8 12.4 156.0 33.3 5743.9 81.8 3164.0 27.3 779.8 74.4 11487.9 139.1 6328.0 38.6 1559.6 108.5 17231.8 176.8 12655.9 65.9 3119.1 155.0 22975.8 210.3 18983.9 82.4 4678.7 186.0 28719.7 241.4 25311.8 101.1 6238.2 209.3 34463.7 270.0 31639.8 116.1 7797.8 232.6 40207.6 297.9 10 100 1000 10000 10 100 Nu Re Therminol 60 (Pr=81.1) Water (Pr=5.93) Air (Pr=0.700) (e) Since Bi # is not important, the functional form reduces to ( ) Nu Re, Pr f = . By observing linear log-log relationship between Nu and Re for a given Pr=constant, we infer that the slope of the linear plot doesnt change with Pr. However, y-intersect of the linear log-log plots is shifted when Pr number changes....
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This note was uploaded on 10/06/2011 for the course MECHANICAL 2.006 taught by Professor Blah during the Spring '08 term at MIT.

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PS2_sol_S08 - Problem Set 2 (2.006, Spring 08) Problem 1...

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