book heat and mass transfer heat-5 ch7-1

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Heat Transfer Su Yongkang

School of Mechanical Engineering

HEAT TRANSFER

CHAPTER 7

External flow

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External Flow: Flat Plate

Topic of the Day

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Where we’ve been ……• General overview of the convection transfer

equations.• Developed the key non-dimensional parameters

used to characterize the boundary layer flow and convective heat and mass transfer.

Where we’re going:• Applications to external flow

– Flat plate Today– Other shapes Next time

Then onto internal flow ……

fk

LhNu

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Differences between external and internal flow

• External flow: Boundary layer develops freely, without constraints

• Internal flow:Boundary layer is constrained and eventually merges

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How this impacts convective heat transfer

• Recall the boundary layer convection equations:

• As you go further from the leading edge, the boundary layer continues to grow. Assuming the surface and freestream T do not change:with increasing distance ‘x’:– Boundary layer thickness, ,

– so

– and

fluid thermal conductivity

wall temperature gradient

TTs

Also

0

y

fs y

Tkq

0

yy

T

sq

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Methods to evaluate convection heat transfer

• Empirical (experimental) analysis– Use experimental measurements in a

controlled lab setting to correlate heat and/or mass transfer in terms of the appropriate non-dimensional parameters

• Theoretical or Analytical approach– Solving of the boundary layer equations for

a particular geometry.– Example:

• Solve for T*• Use evaluate the local Nusselt number, Nux

• Compute local convection coefficient, hx • Use these (integrate) to determine the

average convection coefficient over the entire surface

– Exact solutions possible for simple cases. – Approximate solutions also possible using

an integral method

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Empirical method to obtain heat transfer coefficient

• How to set up an experimental test?

• Let’s say you want to know the heat transfer rate of an airplane wing (with fuel inside) flying at steady conditions………….

• What are the parameters involved?

– Velocity, –wing length,

– Prandtl number, –viscosity,

– Nusselt number,

• Which of these can we control easily?

• Looking for the relation:

Experience has shown the following relation works well:

UT ,

surface wingT

U L

Pr Nu

nmLCNu PrRe

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UT ,inputPower

insulation

L

Empirical method to obtain heat transfer coefficient

• Experimental test setup

• Measure current (hence heat transfer) with various fluids and test conditions for

• Fluid properties are typically evaluated at the mean film temperature

UT ,

2

TT

T sf

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Analytical Solution – Laminar Flow

• Assume:– Steady, incompressible, laminar flow– Constant fluid properties– For flat plate,

• Boundary layer equations

• Blasius developed a similarity solution to the hydrodynamic equations in 1908 based on the stream function, (x,y)

0

y

v

x

u

2

2

y

u

y

uv

x

uu

2

2

y

T

y

Tv

x

Tu

Continuity

Momentum

Energy

UT ,

sT

y

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Analytical Solution – Laminar Flow (Cont’d)

• Define new dependent and independent variables,

• The momentum equation can be rewritten as

• And the boundary conditions are

yu

xv

and

uxu

f/

)(

xuy /

022

2

3

3

d

fdf

d

fd

0)0(0

fd

df

1

d

dfand

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• Blasius solution summary:

• Conclusions from the Blasius solution:

• Solution for the thermal boundary layer:

– For Pr 0.6

– Expressing the local convection coefficient as:

– Then the Local Nusselt number is:

x

xu

x

u Re

x5

Re since but,

5x

u

1 and and x

0 2

Pr *

2

*2

T

fT

31

0η

*

Pr 0.332

T

0η

*

T

x

ukhx

For 0.6 Pr 50

Eq. 7.21

3/12/1 PrRe332.0 xx

x k

xhNu

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• The Average Nusselt number over the whole plate found by integrating:

• Ratio of velocity to thermal boundary layer thickness:

x

1

0

x

xx

x dxhk

x

k

xhuN

y

x

th

For large Pr (oils):

Pr > 1000

y

x

thFor small Pr (liquid metals):

Pr < 0.1Fluid viscosity greater than thermal diffusivity

Fluid viscosity less than thermal diffusivity

Eq. 7.25

3/12/1 PrRe664.0 xxNu

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• Solution for friction factor

• Textbook contains Nusselt number correlations for low Pr (liquid metals) and large Pr (oils)

2/1, Re328.1 xxfC

2/12

,, Re664.0

2/

xxs

xf uC

2/2,

,

u

C xsxf

x

uuxs

332.0,

x

xsxs dxx 0 ,,

1

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Analytical Solution – Turbulent Flow

• For flat plate in turbulent flow (more common)

60 Pr 0.6 Pr Re 0.0296PrStRe

Re 0.37 3154

xx

5-1x

xNu

x

Important point:

– Typically a turbulent boundary layer is preceded by a laminar boundary layer first upstream

need to consider case with mixed boundary layer conditions!

L

xcturb

xc

lamx dxhdxhL

h 1

0

75/1, 10ReRe0592.0

xxxfC

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Analytical Solution – Mixed Boundary Layer

• Integrating

5cx,

85

L1/5L

Lf,

5cx,

85

1/35/4

105Re 10 Re105

Re

1742 -

Re

0.074C

105Re 10 Re105

60 Pr 0.6

871)Pr-Re037.0(

L

L

LLNu

Equations 7.33 and 7.34

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Analytical Solution – Special Cases

• The existence of unheated starting length.

• When the boundary condition is a uniform surface heat flux.For laminar flow,

For turbulent flow,

60Pr 0.6 Pr Re 0.0308 3154x xNu

0.6Pr Pr Re 0.453 3121x xNu

x

ss h

qTxT

)(

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Methodology for a Convection Calculation

• Become immediately cognizant of the flow geometry.

• Specify the appropriate reference temperature and evaluate the fluid properties.

• Calculate the Reynolds number

• Decide whether a local or surface average coefficient is required.

• Select the appropriate correlation.

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Example – Cooling of automobile crankcase

• Given:– Automobile crankcase with approximate

dimensions of 0.6 m long, 0.2 m wide and 0.1 m deep.

– Surface temperature of 350 K– Ambient temperature of 300 K– Vehicle velocity of 30 m/s

• Find:– Heat loss from bottom surface exposed to

air stream

• What other information or assumptions needed?

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Example – Cooling of automobile crankcase (Cont’d)

1. Determine air properties at an average film temperature

2. Calculate Reynolds #

3. Calculate average Nusselt number (mixed b.l.)

4. Average convection coefficient is

5. BOTTOM SURFACE HEAT LOSS:

K 325 2

TTT s

f

Km

W Pr

m

sN

m

kg

23

k

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Example – Cooling of automobile crankcase (Cont’d)

• How to determine the heat loss from the other surfaces?– Assumptions …………..

– Analysis procedure ………

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Example: Cooling air over electronic chips

• Given:Cooling air drawn over electronic devices mounted on board.

• Devices are 4 x 4 mm in size, spacing = 0.25 mm• Find the surface temperature of the fourth device,

assumed uniform surface T.

• Assumptions?

• Solution Method?

T = 27 º C

V = 10 m/sQ = 40 mW each device

“turbulator”

15 mm C L

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Example: Consider atmospheric air at 25 and a ℃velocity of 25 m/s flowing over both surfaces of a 1-m long flat plate that is maintained at 125 . Determine ℃the rate of heat transfer per unit width from the plate for values of the critical Reynolds number corresponding to , , and .510 5105 610

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KEY POINTS THIS SECTION• What key characteristic of external flow

compared to internal flow?• Heat transfer rate generally decreases with

increasing distance from leading edge.• Turbulent convective heat transfer generally

higher than laminar due to mixing effect within boundary layer.

• Experimental tests indicate that heat transfer coefficient will generally vary like:

• Concept of transition Re number.• Difference in boundary layer growth for high

and low Pr number fluids. • General correlation for Nusselt number for flow

over flat plate in laminar, turbulent and mixed flows.

nmL

fL C

k

LhNu Pr Re

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book heat and mass transfer heat-5 ch7-1

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