Solutions_Final_Exams - Problem #1a: (10) From the...

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Problem #1a: (10) From the Maxwell-Stefan equation for a binary gas mixture, derive the following form of Fick’s Law () AA A B A B A x cD x = +−∇ NN N Answer: We start with the Maxwell-Stefan equation in terms of total molar flux 1 AB A AB xx x cD ∇= A B or BA A B A x xc D x =− Adding and subtracting from the left side if this equation gives x N ( ) ABAAA B A x c D +−+ = N x or ( ) A B A B A x cD x = N Problem #1b (15) A fluid of viscosity μ and density ρ is flowing with a mass flow rate w in a smooth horizontal pipe of length L and diameter D as a result of a pressure difference p 0 -p L . The flow is known to be turbulent. The pipe is to be replaced by one of diameter D/2 but with the same length. The same fluid to be pumped at the same flow rate w. Find the pressure difference that will be needed, using the expression for friction factor in a circular tube and Blasius formula. Answer: The expression for the friction factor in a circular tube ( ) 0 2 1 4 12 L z PP D f L u ρ = and the Basius formula for turbulent flow 35 1/4 0.0791 for 2.1 10 Re 10 Re f < < gives 21 2 0 2 51 /4 44 2 2 0.0791 Lz LL w w u f DD D ρρ ρπ π μ −+ ⎛⎞ −= = ⎜⎟ ⎝⎠ / 4 According to this formula, the replacement of the smooth pipe of diameter D by a smooth one of diameter D/2, with all other right-hand variables unchanged, will alter 0 L by a factor of 1/2 27 Problem #2 (15) In many industrial tubular heat exchangers the tube-surface temperature T 0 varies linearly with the bulk fluid temperature T b (i.e., 0 b TT α β = + ). For this common situation h loc and
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h ln may be simply interrelated. Starting from the definition of local heat transfer coefficient for a circular tube of diameter D and length L, () ( ) ( ) 0 loc b loc loc dQ h Ddz T T h Ddz T ππ =− Δ and neglecting the kinetic and potential energy changes, show that ( ) 2 0 1 ˆ 4 loc b p hD d z T T D C u d T ρ ⎛⎞ −= ⎜⎟ ⎝⎠ and therefore that ln ln 0 ( 0 ) 1 ˆ 4 L bb loc p TL T hd z C uD L h T == Δ Answer:
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Solutions_Final_Exams - Problem #1a: (10) From the...

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