ch06 - PROBLEM 6.1 KNOWN Variation of hx with x for laminar...

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PROBLEM 6.1 KNOWN: Variation of h x with x for laminar flow over a flat plate. FIND: Ratio of average coefficient, x h , to local coefficient, h x , at x. SCHEMATIC: ANALYSIS: The average value of h x between 0 and x is x x 0 0 -1/2 x x 1/2 -1/2 x x x 1 C h h dx x dx x x C h 2x 2Cx x h 2h . = = = = = Hence, x x h 2. h = < COMMENTS: Both the local and average coefficients decrease with increasing distance x from the leading edge, as shown in the sketch below.
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PROBLEM 6.2 KNOWN: Variation of local convection coefficient with x for free convection from a vertical heated plate. FIND: Ratio of average to local convection coefficient. SCHEMATIC: ANALYSIS: The average coefficient from 0 to x is x x 0 0 -1/4 x x 3/4 -1/4 x x 1 C h h dx x dx x x 4 C 4 4 h x C x h . 3 x 3 3 = = = = = Hence, x x h 4 . h 3 = < The variations with distance of the local and average convection coefficients are shown in the sketch. COMMENTS: Note that h / h 4/3 x x = is independent of x. Hence the average coefficient for an entire plate of length L is L L 4 h h 3 = , where h L is the local coefficient at x = L. Note also that the average exceeds the local. Why?
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PROBLEM 6.3 KNOWN: Expression for the local heat transfer coefficient of a circular, hot gas jet at T directed normal to a circular plate at T s of radius r o . FIND: Heat transfer rate to the plate by convection. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Flow is axisymmetric about the plate, (3) For h(r), a and b are constants and n -2. ANALYSIS: The convective heat transfer rate to the plate follows from Newton’s law of cooling ( ) ( ) A A conv conv s q dq h r dA T T . = = The local heat transfer coefficient is known to have the form, ( ) n h r a br = + and the differential area on the plate surface is dA 2 r dr. = π Hence, the heat rate is ( ) ( ) ( ) r o 0 o n conv s r 2 n+2 conv s 0 q a br 2 r dr T T a b q 2 T T r r 2 n 2 π π = + = + + ( ) 2 n+2 conv o o s a b q 2 r r T T . 2 n 2 π = + + < COMMENTS: Note the importance of the requirement, n -2. Typically, the radius of the jet is much smaller than that of the plate.
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PROBLEM 6.4 KNOWN: Distribution of local convection coefficient for obstructed parallel flow over a flat plate. FIND: Average heat transfer coefficient and ratio of average to local at the trailing edge. SCHEMATIC: ANALYSIS: The average convection coefficient is ( ) ( ) L L 0 0 2 L x 2 3 2 L 1 1 h h dx 0.7 13.6x 3.4x dx L L 1 h 0.7L 6.8L 1.13L 0.7 6.8L 1.13L L = = + = + = + ( ) ( ) 2 L h 0.7 6.8 3 1.13 9 10.9 W/m K. = + = < The local coefficient at x = 3m is ( ) ( ) 2 L h 0.7 13.6 3 3.4 9 10.9 W/m K. = + = Hence, L L h / h 1.0. = < COMMENTS: The result L L h / h 1.0 = is unique to x = 3m and is a consequence of the existence of a maximum for h x x ± $ . The maximum occurs at x = 2m, where ( ) ( ) 2 2 x x dh / dx 0 and d h / dx 0. = <
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PROBLEM 6.5 KNOWN: Temperature distribution in boundary layer for air flow over a flat plate.
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