A arte da guerra

PROBLEM 6.1 KNOWN: Variation of hx with x for laminar flow over a flat plate. FIND: Ratio of average coefficient, h x , to local coefficient, hx, at x. SCHEMATIC:

ANALYSIS: The average value of hx between 0 and x is hx = hx hx Hence, 1 x C x ∫ h x dx = ∫ x -1/2dx x 0 x 0 C 1/2 = 2x = 2Cx -1/2 x = 2h x . hx = 2. hx

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COMMENTS: Both the local and average coefficients decrease with increasing distance x from the leading edge, as shown in the sketch below.

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 1 x C x -1/4 h x = ∫ h x dx = ∫ x dx x 0 x 0 4 C 3/4 4 4 hx = x = C x -1/4 = h x . 3 x 3 3 Hence, hx 4 = . hx 3

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The variations with distance of the local and average convection coefficients are shown in the sketch.

COMMENTS: Note that h x / h x = 4 / 3 is independent of x. Hence the average coefficient 4 for an entire plate of length L is h L = h L , where hL is the local coefficient at x = L. Note 3 also that the average exceeds the local. Why?

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 Ts of radius ro. 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 q conv = ∫ dq conv = ∫ h ( r ) ⋅ dA ⋅ ( T∞ − Ts ).
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The local heat transfer coefficient is known to have the form, h ( r ) = a + br n and the differential area on the plate surface is dA = 2π r dr.

Hence, the heat rate is q conv = ∫ ro 0

(a + brn ) ⋅ 2π r dr ⋅ (T∞ − Ts ) r b n+2  o a q conv = 2π ( T∞ − Ts )  r 2 + r  n+2 2 0 b n+2  a 2 q conv =