boudarylayer(undisturbed flow encounters the plate)

8/14/2019 Boudarylayer(undisturbed flow encounters the plate)

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as well. By analogy. we can expect that h(x) and Nu(x) will also decrease withdistance along the plate.

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

u [m/s]

y[m]

x=1mx=2mx=3m

Figure 2: Velocity profiles at three locations along the plate plotted on a sinlgeaxis.

The rate of heat transfer will be highest where the gradients are highest,and clearly this occurs at the leading edge. From our dimensional analysis, weexpect

h cfPrn (3)

since

cf = f(x,Re) (4)

Nu = f(x,Re,Pr) (5)

Example

Consider airflow over a plate of length 1m. Assume transition to turbulenceoccurs at xc = 0.5m, with Rec = 5 105. Given hlam(x) = Clamx0.5 andhturb(x) = Cturbx

0.2, with Clam = 8.845W/(m3/2K) and Cturb = 49.75W/(m

1.8K).

1. Using air at 350 K determine U

2. Develop an expression for the average convection coefficient as a functionof distance from the leading edge, for the laminar region.

3. Develop an expression for the average convection coefficient, hturb(x) as afunction of distance from the leading edge, for the turbulent region.

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4. Plot the local and average coefficients for (0 x L)

Solution

a)

Assumptions: hlam(x) is as given, constant fluid properties.Properties: Air @350K, Table A.4 = 20.92 106m2/s

Rex,c =Uxc

(6)

U =Rex,c

xc=

5 10520.92 1060.5

= 20.9[m/s] (7)

b)

hlam(x) =1

x

x0

hlam(x)dx

=1

xClam(2)x

1/2

= 2Clamx1/2

At any point, the average convection coefficient up to that point is twice thelocal value.

c)

The average convection coefficient in the turbulent region must account for thelaminar portion at the beginning of the plate.

hturb(x) =1

xc

xc0

hlam(x)dx +1

x xc

xxc

hturb(x)dx

=2Clamx

0.5c

xc+

Cturbx xc

x0.8

0.8

x

xc

= 2Clamx1/2

+ 1.25Cturb

x0.2 x

0.8c

x

d)

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0 0.5 1 1.510

20

30

40

50

60

70

80

x [m]

h[W/m2K

Variation of Average and local h

hlamAverage(h

lam)h

turbAverage(h

turb)

Figure 3: Convection coefficients along the plate.

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boudarylayer(undisturbed flow encounters the plate)

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