Heat Transfer Su Yongkang
School of Mechanical Engineering
# 1
HEAT TRANSFER
CHAPTER 8
Internal flow
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
Internal Flow Heat Transfer
Where we’ve been ……
Introduction to internal flow,boundary layer
growth,entry effects
Where we’re going:
Developing heat transfer coefficient
relationships and correlations for internal flow
ro
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
Internal Flow Heat Transfer
KEY POINTS THIS LECTURE
Energy balance for internal flow in a tube
Temperature and heat transfer relations for two
cases:
– Constant surface heat flux
– Constant surface temperature
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
Basic concepts—thermal considerations
1,The mean temperature
For the internal energy
So
For incompressible flow in a circular tube,
ro
cA cvt T d AucE mvt TcmE
and
vcm
A cv
v
A cv
m cAu
T d Auc
cm
T d Auc
T cc

002
0
2 r
m
m u T r d rruT
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
Basic concepts—thermal considerations
2,Newton’s law of cooling
Here,the mean T plays the same role as the free
stream for external flows.
3,Fully developed conditions
because of heat transfer,T(r) is continuously
changing with x,
can fully developed conditions be reached
For thermally fully developed
Although the temperature profile T(r) changes
with x,the relative shape of the profile no longer
changes.
)( mss TThq
T
Not constant!
0
)()(
),()(
,

tfdms
s
xTxT
xrTxT
x
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
Basic concepts—thermal considerations
In the thermally fully developed flow of a fluid
with constant properties,the local convection
coefficient is a constant,independent of x.
For the special case of uniform surface heat flux
For the case of constant surface temperature
)( xfh?
tfd
m
tfd dx
dT
x
T
,,
tc o n sq s t a n
Axial T
gradient
Independent of
radial location
tfd
m
ms
s
tfd dx
dT
TT
TT
x
T
,,)(
)(

tco n sT s t an?
Depends on the
radial coordinate
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
Example,Velocity and temperature profiles
for laminar flow in a tube of radius
have the form
with units of m/s and K,respectively,
Determine the corresponding value of the mean
(or bulk) temperature,,at this axial position.
mmro 10?
mT

42
2
/8.18/0.758.3 4 4)(
/11.0)(
oo
o
rrrrrT
rrru

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
Review,Energy Balance Analysis
x x + dx
advinE,? advoutE,?
convectionE?
Energy Balance
Change in energy in the control volume =
energy input – work out + energy in by advection
– energy carried out by advection (flow)
or
Two Special Cases:
1,Constant surface heat flux
2,Constant surface temperature
dxqdxPqE
TcmE
TcmE
co n vco n vco n v
dxxmpa d vo u t
xmpa d vin

,,
,,

0
0,,
ms
pp
c on vm
mpc on v
dxxmpxmpc on v
TTh
cm
P
cm
Pq
dx
dT
dTcmdxPq
TcmTcmdxPq

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Case 1,Constant Surface Heat Flux
Example application
– Electrical heater element around a pipe
Recognize that,
From the energy balance equation
if q” is constant
If q” is a known function
of x instead,must
integrate the above to
obtain Tm,x
)( xfq s
T )( im,xcm PqxT
p
s
m?

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
Example,Consider flow in a circular tube,Within
the test section length (between 1 and 2) a constant
heat flux is maintained,
For the following two cases,sketch the surface
temperature and the fluid mean temperature
as a function of distance along the test
section x,In case A flow is hydrodynamically and
thermally fully developed,In case B flow is not
developed.
sq?
)(xTs
)(xTm
Solution:
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Example,Pre-heating Water
Given:
An industrial process requires 0.2 kg/s of pre-
heated water,The average tap water
temperature during winter is known to be 10o C,
You are considering using insulated electrical
heater elements wrapped around the water pipe
to heat the water.
– Electrical heaters provide 5 kW / m2 to the
pipe
– Existing pipe is 0.05m I.D.,0.055m O.D.
– Internal h = 500 W / m2K
– Available pipe length = 10 m
1,How hot can the outlet water temperature get
in these conditions?
2,What is wall temperature at exit?
q?
C10inT CoutT
inpco n vxm Tcm
PqT x
,

Heat Transfer Su Yongkang
School of Mechanical Engineering
# 14
Example (Cont’d)
Solution:
At the inlet condition,
Perimeter of pipe:
Calculations:
If the internal convection coefficient =
what is the inner wall T at the exit?
q?
C10inT CoutT
inpco n vxm Tcm
PqT x
,

Km
W
2500
05.0 DP
)(,o u tw a l loc o n v TThq
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 15
Case 2,Constant Surface Temperature
Example application
– Steam condensation on the outer surface of the
tube
Start with the same general energy balance
equation
Separate variables and integrate from inlet to outlet
0
1 -
L
p
PL h d x
m c L

Lh
hcm LPTT
pi
o
- ln

Th
cm
P
dx
Td
dx
dT
TTTD e f i n e
p
m
ms

)(
i
o
T
T
ln

h
cm
LP
T
T
pim
om
-e x p
T
T
,s
,s
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 16
Constant Surface Temperature
Temperature asymptotically approaches surface
temperature Ts as you go downstream
At any location ‘x’:
mT
sT
T
x
mT
sT
T
x
is TT if? is TT if?
hcm xPTT xTT
pims
ms
-e x p )(
,?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 17
Log Mean Temperature Difference
Define:
Log mean temperature difference (LMTD) as:
Convective heat transfer is defined by:
and
then

oip osispiopco n v TTcm
TTTTcmTTcmq

ln
-
- ln
i
o
p
pi
o
T
T
hLPcmh
cm
LP
T
T

-
ln
c onv i oo
i
P L hq T T
T
T

)/ln ( io
io
lm TT
TTT

lmTPLh?
lmsc o n v TAhq
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 18
Overall Heat Transfer Coefficient,
Recall from the discussion in conduction heat
transfer,in general:
Put energy balance equation in terms of ambient
temperature,T?
U
TAUq sco n v
ir
orim
i Tmh
,,?
omTm,,Tho,
Define

p
s
im
om
i
o
cm
AU
TT
TT
T
T
-e x p
,
,
t o tR
lmlmsc o n v TTAUq
s
to t AUR
1?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 19
Back to Pre-heating Water Example
Revised Problem Statement:
You decide to also investigate using available
steam to heat the water.
– Assume pipe wall temperature at 100 o C
(condensing steam T)
Find:
How long would the pre-heating pipe length have
to be to give an outlet temperature = 50o C?
Solution method
C100sT
C10inTL en g th C50outT
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 20
Pre-heating Water Example – Part 2
Solution:
Fluid properties at a mean T = 30oC
Reynolds number:
For turbulent flow,heating mode
Log mean temperature difference
C100sT
C10inTL en g th C50outT
5,8 3Pr ;6 1 3.0;1 0 0 0 ;108 5 5 ;1 7 9.4 326

mK
Wk
m
kg
m
sN
k g K
kJc
p
Tu r b u l e n t ! 5956)05.0)(10855( )/2.0(4 4Re 6 mskgDmD
Km
W
5 9 8
m05.0
W / m K0,6 1 3
4 8,8
8.48
)83.5()0,0 2 3 ( 5 9 5 6 PrRe 0 2 3.0
2
4.00,84.05/4

h
D k
h
Nu
Nu
D
DD
C68
101 0 0
501 0 0ln
C)101 0 0(C)501 0 0(

LMTL M T D
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 21
Pre-heating Water Example – Part 2
Solution (Cont’d):
Compute the pipe wall surface area required
Required length to get this area is:
What would be the pro and cons for choosing
electrical heater versus heating with steam in
this case?
C100sT
C10inTL en g th C50outT

2
2
m 0,8 2 2
68
Km
W
598 1050179.42.0
Δ

s
s
LMsino u tpc o n v
A
C AC
kg K
kJ
s
kg
T AhTTcmq

,0 5 m0 m 0,8 2 2 2 LD LA s
m 5.2L?
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 22