# 1

Heat Transfer Su Yongkang

School of Mechanical Engineering

HEAT TRANSFER

Final Review

# 2

Heat Transfer Su Yongkang

School of Mechanical Engineering

Final Review Session

# 3

Heat Transfer Su Yongkang

School of Mechanical Engineering

Viscous Flow

• The Navier-Stokes EquationsNonlinear, second order, partial differential equations.

• Couette Flow, Poiseuille Flow.

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

z

w

y

w

x

wg

z

p

z

ww

y

wv

x

wu

t

w

z

v

y

v

x

vg

y

p

z

vw

y

vv

x

vu

t

v

z

u

y

u

x

ug

x

p

z

uw

y

uv

x

uu

t

u

z

y

x

0

z

w

y

v

x

u

# 4

Heat Transfer Su Yongkang

School of Mechanical Engineering

Convection

• Basic heat transfer equation

• Primary issue is in getting convective heat transfer coefficient, h

• h relates to the conduction into the fluid at the wall

)( TTAhq ss h average heat transfer coefficient

L

As

sdxh

LhdAh

Ah

s 0

1 :unit widthfor or,

1

TT

y

Tk

hs

yf

x0

-

# 5

Heat Transfer Su Yongkang

School of Mechanical Engineering

Convection Heat Transfer Correlations

• Key is to fully understand the type of problem and then make sure you apply the appropriate convective heat transfer coefficient correlation

External FlowFor laminar flow over flat plate

For mixed laminar and turbulent flow over flat plate

0dx

dP

UT ,

sT

y

31

21

x Pr Re 0.332

k

xhNu x

x3

12

1

x Pr Re 0.664

k

xhuN x

x

L

xcturb

xc

lamx dxhdxhL

h 1

0

7.41 Eq.

105Re 10 Re105

60 Pr 0.6

Pr 871Re 0.037

5cx,

85

3154L

L

LNu

# 6

Heat Transfer Su Yongkang

School of Mechanical Engineering

External Convection Flow

For flow over cylinderOverall Average Nusselt number

Table 7.2 has constants C and m as f(Re)

For flow over sphere

For falling liquid drop

4131

Pr

Pr Pr Re

s

mDD C

k

DhNu

414.03221 Pr)Re 0.06 Re (0.4 2

s

DDD k

DhNu

3121 Pr Re 0.6 2 DDNu

# 7

Heat Transfer Su Yongkang

School of Mechanical Engineering

Convection with Internal Flow

• Main difference is the constrained boundary layer

• Different entry length for laminar and turbulent flow

• Compare external and internal flow:

– External flow:Reference temperature: T is constant

– Internal flow:Reference temperature: Tm will change if heat transfer is occurring!

• Tm increases if heating occurs (Ts > Tm )

• Tm decreases if cooling occurs (Ts < Tm )

ro

# 8

Heat Transfer Su Yongkang

School of Mechanical Engineering

Internal Flow (Cont’d)

• For constant heat flux:

• For constant wall temperature

• Sections 8.4 and 8.5 contain correlation equations for Nusselt number

)(xTs

)(xTm

thermalfdx ,

T

x

mT

sT

T

x

mT

sTT

x

is TT if is TT if

LMsconv T h Aq

inp

convxm T

cm

qT x

,

# 9

Heat Transfer Su Yongkang

School of Mechanical Engineering

Free (Natural) Convection

• Grashof number in natural convection is analogous to the Reynolds number in forced convection

Unstable,Bulk fluid motion

Stable,No fluid motion

forces Viscous

forcesBuoyancy

2

3

LTTg

Gr sL

1Re2

L

LGr1

Re2

L

LGr Natural convection dominates

Natural convection can be neglected

# 10

Heat Transfer Su Yongkang

School of Mechanical Engineering

Free (Natural) Convection

Rayleigh number: For relative magnitude ofbuoyancy and viscous forces

• Review the basic equations for different potential cases, such as vertical plates, vertical cylinders, horizontal plates (heated and cooled)

• For horizontal plates, discuss the equations 9.30-9.32. (P513)

• Please refer to problem 9.34.

Pr xx GrRa

For vertical surface, transition to turbulence at Rax 109

# 11

Heat Transfer Su Yongkang

School of Mechanical Engineering

Heat Exchangers

• Two basic methods discussed:1. LMTD Method

2. -NTU Method

outBT ,

side) (shell ,inBT

side) (tube ,inAT

outAT ,

Example:Shell and Tube:Cross-counter Flow

LMTD

i

o

inout TUA

T

TTT

UAq

ln

icih TTCqor

qq

,,min

max

:

icih TTCqwhereq

q

,,minmax

max

:

min

, NTUC

UAHXoverall

rCNTUf ,

1 C C rmax

minr

C

C

# 12

Heat Transfer Su Yongkang

School of Mechanical Engineering

Discussion on the U

• Equation 11.5

• For the unfinned, concentric, tubular heat exchangers.

• When the inner tube surface area is the reference calculating area.

• When the inner tube surface area is the reference calculating area.

ooo

ofio

i

if

ii

ooii

AhA

R

kL

DD

A

R

Ah

AUAUUA

1

2

)/ln(1

111

,,

oo

i

o

iofi

ioif

ii Ah

A

A

ARA

kL

DDR

hU

,

, 2

)/ln(11

ii

o

i

oifo

ioof

oo Ah

A

A

ARA

kL

DDR

hU

,

, 2

)/ln(11

Example 11.1Notice!

# 13

Heat Transfer Su Yongkang

School of Mechanical Engineering

Discussion on the problems