CHAPTER 7 Heat Transfer

5/19/2018 CHAPTER 7 Heat Transfer

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EXTERNAL FORCEDCONVECTION

Prepared byNURHASLINA CHE RADZI

FKK, UITM


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Flat Plate in Parallel Flow

1) Laminar Flow over an isothermal Plate


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Boundary layer thickness as that value of y for which (u/u) = 0.99

= 5.0 = 5x , Rex = ux

(u/x) Rex

From the above equation, increases with increasing x and but

decreases with increasing u The larger free stream velocity, the thinner the boundary layer

The wall shear stress may be expressed as

s = u = u (u/x) d2f

y y=0 d2 =0

value of , f , df/dand d2f/d2 can get from Table 7.1

s = 0.332 u ( u /x)


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The local friction is then

Cf,x = s,x = 0.664 Rex-1/2

u2/2

For problems involving the formation of boundary layer, a non-dimensional form for the heat transfer coefficient is :

- Local Nusselt number

- the ratio of the velocity to thermal boundary layer thickness is

Pr),Re*,x(f=Nu xx

0.6Pr,PrRe332.03/12/1 x

xx

k

xhNu

31

rt

P


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- From the foregoing local result , average boundary layerparameters may be determined . With the average friction

coefficient defined as

- The average heat transfer coefficient for laminar flow is

x

0x,sx,s

2

x,sx,f

dxx

1

where

2/u

C

21xx,f Re328.1=C-

( ) x

0 21

21

31x

0xx

x

dx/uPr

x

k332.0=dxh

x

1=h

hx = 2hx


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Average Nusselt number:

A single correlating equation, which applies for all Prandtl numbersfor laminar flow over an isothermal plate, the local convectioncoefficient may be obtained from (Churchill and Ozoe):

Nux = 0.3387 Rex1/2Pr1/3 , Pex 100[1 + (0.0468/Pr)2/3]1/4 Pex= Pr Rex

Nux = 2Nux

0.6Pr,PrRe664.03/12/1

xx

x k

xh

Nu

Fluid properties are usually evaluated at the film temperature:

2

TT

T sf


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2) Turbulent Flow over an isothermal Plate

- Local friction coefficient is

Cf,x = 0.0592 Rex-1/5 , Rex,c Rex 10

8

- The velocity boundary layer thickness is

= 0.37 x Rex-1/5

- Local Nusselt number for turbulent flow (Chilton-Colburn) is

60Pr0.6,PrRe0296.03/15/4 xxNu

Fluid properties are usually evaluated at the film temperature:

2

TT

T sf


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3) Mixed Boundary Layer Conditions

- In the mixed boundary layer, average convection coefficient may

be expressed asNuL = hLL = (0.037 ReL4/5A) Pr1/3

k

0.6 Pr 60 Rex,c = critical Reynolds number

Rex,c Rex,L 108 A = 0.037 Rex,c

4/50.664 Rex,c1/2

- Average friction coefficient for mixed boundary layer

Cf,L = 0.074 ReL-1/5(2A/Rel) [Rex,c ReL 10

8]

- For completely turbulent boundary layer

- Rex,c= 0- A = 0

- For a transition Reynolds number- Rex,c= 5 x 10

5

- A = 871

Fluid properties are

usually evaluated at the

film temperature:

2

TT

T sf


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From the figure above, velocity boundary layer growth begins at x=0

Thermal boundary layer development begins at x =

No heat transfer for 0 x

For laminar flow:

Nux = Nux =0

[1 (/x)3/4]1/3 For turbulent flow

Nux = Nux =0

[1 (/x)9/10]1/9 For a plate of total length L, with laminar or turbulent flow:

NuL = NuL =0 L [1 (/L)(P + 1)(P + 2)]P/(P +1)

L

P = 2 for laminar flow NuL

= hL , L = L -

P = 8 for turbulent flow k

u , T

Ts = T

tqsTs > T

4) Unheated Starting Length ( Ts= T)

X = L


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5) Flat Plates with constant Heat Flux Conditions

For laminar flow:

Nux= 0.453 Rex1/2Pr1/3 , Pr 0.6

For turbulent flow:

Nux= 0.0308 Rex4/5Pr1/3 , 0.6 Pr 60

If the heat flux is known, the convection coefficient may be used todetermine the local surface temperature

Ts(x) = T+ qs

hx

Average surface temperature

(Ts- T) = qsL , NuL= 0.680 ReL1/2Pr1/3

k NuL


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PROBLEM 7.1

Consider the following fluids at a film temperature of 300k inparallel flow over a flat plate with velocity of 1 m/s. atmospheric air,water, engine oil, and mercury.

a) For each fluid, determine the velocity and thermal boundary layerthickness at a distance of 40mm from the leading edge.

b) If a flow is laminar, the following expressions may be used tocompute and trespectively

= 5x t = Rex

1/2 Pr1/3


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PROBLEM 7.2

Engine oil at 100c and a velocity of 0.1m/s flows over bothsurfaces of a 1m flat plate maintained at 20c.

Determine:

a) The velocity and thermal boundary layer thickness at the trailingedge

b) The local heat flux and surface shear stress at the trailing edge

c) The total drag force and heat transfer per unit width of the plate


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PROBLEM 7.10

Consider atmospheric air at 25c and a velocity of 25m/sflowing over both surface of a 1m long flat plate that ismaintained at 125c. Determine the rate of heat transfer perunit width from the plate for values of the critical Reynolds

number corresponding to 105, 5 x 105and 106


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PROBLEM 7.15

Air at pressure of 1 atm and a temperature of 50c is inparallel flow over the top surface of a flat plate that is heatedto a uniform temp. of 100c. The plate has or length of0.20m and a width of 0.10m. The Reynoldsnumber based on

the plate length is 40,000. What is the rate of heat transferfrom the plate to the air? If the free stream velocity of the airdoubled and the pressure is increased to 10 atm, what is therate of heat transfer?


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PROBLEM 7.11

Consider a flat plate subjects to parallel flow (top and bottom)characterized by u= 5m/s, T = 20c

a) Determine the average convective heat transfer coefficient,

convective heat transfer rate and drag force associated with a L =2m long, w = 2m wide flat plate for air flow and surface temp. ofTs= 50c and 80c

b) Determine the average convective heat transfer coefficient,

convective heat transfer rate and drag force witk a L = 0.1m long,w = 0.1m wide flat plate for water flow and surface temp. of Ts=50c and 80c


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Flow around Cylinders and Spheres

Flow around cylinders and spheres is characterized by boundarylayer development and separation.

Heat transfer coefficients are strongly influenced by the nature ofboundary layer development at the surface.


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For the circular cylinder, the characteristic length is the diameter

and the Reynolds number is defined as

Red = vD = vD

For laminar boundary layer :

Red 2 x 105

= 80c

For transition boundary layer :

Red 2 x 105

= 140c


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CROSSFLOW AROUND CYLINDERS1) Hilpert Correlation

Can be used for cross flow around other non-circular shapes

See Table 7.2 for values of C and m

NuD = hD CReDmPr1/3

k

2) Zukaukas Correlation

C and m are listed in Table 7.4

3) Churchill and Bernstein correlation, for all ReDand Pr > 0.2

Drag Coefficient:

CD = FD , FD = drag force

Af(v2/2) Af = cylinder frontal area

6D

4/1

10Re1500,Pr0.7PrPrPrRe

s

nmDD C

kDhNu

5/48/5

4/13/2

3/12/1

000,282

Re1

Pr/4.01

PrRe62.03.0

DDDNu


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CROSSFLOW AROUND SPHERES1) Whitaker correlation:

All properties are evaluated at Texcept swhich is evaluated at Ts

2) Correlation by Ranz and Marshall for heat transfer from freelyfalling liquid drops:

3) At ReD = 0 , equations above reduce to:

For Whitaker correlation:0.71 Pr 380

3.5 ReD 7.6 x 104

1.0 (/s) 3.2

Drag coefficient:

CD = 24/ReD , ReD 0.5

4/1

4.03/22/1Pr)Re06.0Re4.0(2

s

DDDNu

3/12/1 PrRe6.02 DDNu

2DNu


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FLOW ACROSS BANKS OF TUBES Tube arrangements in bank

Aligned Staggered

Grimison correlation

NuD = C1ReDm, max NL 10 , Pr = 0.7

2000 ReD, max 40,000

C1and m are listed in Table 7.5

ReD, max = vmaxD , vmax = ST ST- D

NuD = 1.13 C1ReDm, maxPr

1/3 NL 10 , Pr 0.7

2000 ReD, max 40,000

All properties are evaluated at film temperature

v,T

SL

ST A1

D v,T

SL

ST A1A2

D

SD

v


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If NL< 10

NuD = C2NuD NL 10

C2is given Table 7.6

Zukauskas correlation

NuD = C ReDm, maxPr

0.36(Pr/Prs)1/4

NL 20 C and m are listed in Table 7.7 . All properties0.7 Pr 500 except Prsare evaluated at the arithmetic

1000 ReD , max 2 x 106 mean of the fluid inlet and outlet temperature

If NL

CHAPTER 7 Heat Transfer

Documents

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formation of boundary

arel rex

uxux rex

velocity boundary layer

laminar flow nul

laminar flow isx0x

increasing x