Heat Transfer Su Yongkang
School of Mechanical Engineering
# 1
HEAT TRANSFER
Final Review
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
Final Review Session
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
Viscous Flow
The Navier-Stokes Equations
Nonlinear,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
w
g
z
p
z
w
w
y
w
v
x
w
u
t
w
z
v
y
v
x
v
g
y
p
z
v
w
y
v
v
x
v
u
t
v
z
u
y
u
x
u
g
x
p
z
u
w
y
u
v
x
u
u
t
u
z
y
x
0 zwyvxu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
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
A ss
dxhLhdAhAh
s 0
1,u n i t w i d t hf o r o r,1
TT
y
Tk
h
s
y
f
x
0
-
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
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 Flow
For laminar flow over flat plate
For mixed laminar and turbulent flow over flat plate
0?dxdP
UT,
sT
y
3121x Pr Re 0,3 3 2 k xhNu xx 3121x Pr Re 0,6 6 4 k xhuN xx
L
xc t u r b
xc
l a mx dxhdxhLh 1 0
7,4 1 E q,
105Re 10 Re105
60 P r 0,6
Pr 871Re 0,0 3 7
5
cx,
85
3154
L
L
LNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
External Convection Flow
For flow over cylinder
Overall 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 0,0 6 Re ( 0,4 2
sDDD k
DhNu
3121 Pr Re 0,6 2 DDNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
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
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
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
sT
T
x
is TT if? is TT if?
LMsco n v T h Aq
inpco n vxm Tcm
qT x
,
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
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
f o r c e s V i s c o u s
f o r c e sB u o y a n c y
2
3
LTTgGr s
L
1Re 2
L
LGr 1
Re 2L
LGr
Natural
convection
dominates
Natural
convection can
be neglected
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Free (Natural) Convection
Rayleigh number,For relative magnitude of
buoyancy 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
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
Heat Exchangers
Two basic methods discussed:
1,LMTD Method
2,?-NTU Method
outBT,
s id e ) ( s h e ll,inBT
s id e ) ( tu b e,inAT
outAT,
Example:
Shell and Tube:
Cross-counter Flow
L M T D
i
o
ino u t TUA
T
T
TTUAq
ln
icih TTCq
or
qq
,,m in
m ax
:
icih TTCqw e r e
q
q
,,m inm a x
m a x
,
m in
,N TU
C
UA HXo v e r a l lrCN T Uf,
1 C r
m a x
m inr CC
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
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
)/l n (1
111
,,?
oo
i
o
iof
i
io
if
ii Ah
A
A
ARA
kL
DDR
hU?
,
,2
)/l n(11
ii
o
i
oif
o
io
of
oo Ah
A
A
ARA
kL
DDR
hU?
,
,2
)/l n (11
Example 11.1 Notice!
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Discussion on the problems
School of Mechanical Engineering
# 1
HEAT TRANSFER
Final Review
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 2
Final Review Session
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 3
Viscous Flow
The Navier-Stokes Equations
Nonlinear,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
w
g
z
p
z
w
w
y
w
v
x
w
u
t
w
z
v
y
v
x
v
g
y
p
z
v
w
y
v
v
x
v
u
t
v
z
u
y
u
x
u
g
x
p
z
u
w
y
u
v
x
u
u
t
u
z
y
x
0 zwyvxu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 4
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
A ss
dxhLhdAhAh
s 0
1,u n i t w i d t hf o r o r,1
TT
y
Tk
h
s
y
f
x
0
-
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 5
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 Flow
For laminar flow over flat plate
For mixed laminar and turbulent flow over flat plate
0?dxdP
UT,
sT
y
3121x Pr Re 0,3 3 2 k xhNu xx 3121x Pr Re 0,6 6 4 k xhuN xx
L
xc t u r b
xc
l a mx dxhdxhLh 1 0
7,4 1 E q,
105Re 10 Re105
60 P r 0,6
Pr 871Re 0,0 3 7
5
cx,
85
3154
L
L
LNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 6
External Convection Flow
For flow over cylinder
Overall 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 0,0 6 Re ( 0,4 2
sDDD k
DhNu
3121 Pr Re 0,6 2 DDNu
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 7
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
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 8
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
sT
T
x
is TT if? is TT if?
LMsco n v T h Aq
inpco n vxm Tcm
qT x
,
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 9
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
f o r c e s V i s c o u s
f o r c e sB u o y a n c y
2
3
LTTgGr s
L
1Re 2
L
LGr 1
Re 2L
LGr
Natural
convection
dominates
Natural
convection can
be neglected
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 10
Free (Natural) Convection
Rayleigh number,For relative magnitude of
buoyancy 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
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 11
Heat Exchangers
Two basic methods discussed:
1,LMTD Method
2,?-NTU Method
outBT,
s id e ) ( s h e ll,inBT
s id e ) ( tu b e,inAT
outAT,
Example:
Shell and Tube:
Cross-counter Flow
L M T D
i
o
ino u t TUA
T
T
TTUAq
ln
icih TTCq
or
,,m in
m ax
:
icih TTCqw e r e
q
q
,,m inm a x
m a x
,
m in
,N TU
C
UA HXo v e r a l lrCN T Uf,
1 C r
m a x
m inr CC
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 12
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
)/l n (1
111
,,?
oo
i
o
iof
i
io
if
ii Ah
A
A
ARA
kL
DDR
hU?
,
,2
)/l n(11
ii
o
i
oif
o
io
of
oo Ah
A
A
ARA
kL
DDR
hU?
,
,2
)/l n (11
Example 11.1 Notice!
Heat Transfer Su Yongkang
School of Mechanical Engineering
# 13
Discussion on the problems
