PROBLEM 6.9 KNOWN: Variation of local convection coefficient with distance x from a heated plate with a uniform temperature Ts. FIND: (a) An expression for the average coefficient 12h for the section of length (x2 - x1) in terms of C, x1 and x2, and (b) An expression for 12h in terms of x1 and x2, and the average coefficients 1h and

2h , corresponding to lengths x1 and x2, respectively. SCHEMATIC:

hx = Cx-1/2dq

ASSUMPTIONS: (1) Laminar flow over a plate with uniform surface temperature, Ts, and (2) Spatial

variation of local coefficient is of the form 1/ 2xh Cx= , where C is a constant.

ANALYSIS: (a) The heat transfer rate per unit width from a longitudinal section, x2 - x1, can be expressed as ( )(12 12 2 1 sq h x x T T ) = (1) where 12h is the average coefficient for the section of length (x2 - x1). The heat rate can also be written in terms of the local coefficient, Eq. (6.11), as

(2) ( ) ( )21 1

x x12 x s s xx

q h dx T T T T h = = 2x dxCombining Eq. (1) and (2),

( ) 21x

12 xx2 1

1hx x

= h dx (3) and substituting for the form of the local coefficient, 1/ 2xh Cx

= , find that

( )2

21

1

x 1/ 2 1/ 21/ 2x 1/ 2 2 112 x2 1 2 1 2 1x

x x1 C xh Cx dx 2Cx x x x 1/ 2 x x

= = = (4)< (b) The heat rate, given as Eq. (1), can also be expressed as ( ) ( )12 2 2 s 1 1 sq h x T T h x T T = (5) which is the difference between the heat rate for the plate over the section (0 - x2) and over the section (0 - x1). Combining Eqs. (1) and (5), find,

2 2 1 1122 1

h x h xhx x

= (6)< COMMENTS: (1) Note that, from Eq. 6.6,

x x 1/ 2 1/ 2

x x0 01 1h h dx Cx dx 2Cxx x

= = = (7) or xh = 2hx. Substituting Eq. (7) into Eq. (6), see that the result is the same as Eq. (4).

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