# Exercícios Resolvidos Incropera capítulos 6 ao 9

(Parte 1 de 11)

PROBLEM 6.1

KNOWN: Variation of hx with x for laminar flow over a flat plate.

FIND: Ratio of average coefficient, xh, to local coefficient, hx, at x. SCHEMATIC:

ANALYSIS: The average value of hx between 0 and x is

1Ch h dx x dx x

Ch2 x 2Cx x

Hence, 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.

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

1Ch h dx x dx x

Hence, x

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

COMMENTS: Note that h/h4/3xx= is independent of x. Hence the average coefficient for an entire plate of length L is LL4h h3 =, where hL is the local coefficient at x = L. Note 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.

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

The local heat transfer coefficient is known to have the form,

()nhra br=+ and the differential area on the plate surface is dA2 r dr.=π Hence, the heat rate is n conv s

r 2n +2 conv s qabr2 r drTT abq2 T T r r 2n 2

()2n +2 convoosabq2rr T.

COMMENTS: Note the importance of the requirement, n ≠ -2. Typically, the radius of the jet is much smaller than that of the plate.

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

2 Lx

11h h dx 0.7 13.6x 3.4x dx L

COMMENTS: The result LLh/h1.0= is unique to x = 3m and is a consequence of the existence of a maximum for hxx . The maximum occurs at x = 2m, where

PROBLEM 6.5

KNOWN: Temperature distribution in boundary layer for air flow over a flat plate. FIND: Variation of local convection coefficient along the plate and value of average coefficient. SCHEMATIC:

ANALYSIS: From Eq. 6.17,

where Ts = T(x,0) = 90°C. Evaluating k at the arithmetic mean of the freestream and surface temperatures, T = (20 + 90)°C/2 = 55°C = 328 K, Table A.4 yields k = 0.0284 W/m⋅K. Hence, with

and the convection coefficient increases linearly with x.

The average coefficient over the range 0 ≤ x ≤ 5 m is

PROBLEM 6.6

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.

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

Spatial variation of local coefficient is of the form 1/2xhCx−=, where C is a constant.

ANALYSIS: (a) The heat transfer rate per unit width from a longitudinal section, x2 - x1, can be expressed as

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. (6.3), as

Combining Eq. (1) and (2),

xx1C xhC x dx 2C x x 1/2 x x

(b) The heat rate, given as Eq. (1), can also be expressed as

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,

(Parte 1 de 11)