HEAT
TRANSFER
Final Review
Heat Transfer
#1
Su Yongkang
School of Mechanical Engineering
Final Review Session
Heat Transfer
#2
Su Yongkang
School of Mechanical Engineering
Viscous Flow
• The Navier-Stokes Equations
Nonlinear, second order, partial differential equations.
  2u  2u  2u 
 u
u
u
u 
p
   u  v  w     g x    2  2  2 
x
y
z 
x
y
z 
 t
 x
  2v  2v  2v 
 v
v
v
v 
p
   u  v  w     g y    2  2  2 
x
y
z 
y
y
z 
 t
 x
 2w 2w 2w 
 w
w
w
w 
p
   u
v
 w     g z    2  2  2 
x
y
z 
z
y
z 
 t
 x
u v w
 
0
x y z
• Couette Flow, Poiseuille Flow.
Heat Transfer
#3
Su Yongkang
School of Mechanical Engineering
Convection
• Basic heat transfer equation
h  average heat
q  h As (Ts  T )
transfer coefficient
• Primary issue is in getting convective heat
transfer coefficient, h
1
h
 h dAs
As As
1 L
or, for unit width : h 
 h dx
L 0
• h relates to the conduction into the fluid at the
wall
T
-kf
y y  0
hx 
Ts  T 
Heat Transfer
#4
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 Flow
For laminar flow over flat plate
dP
0
dx
y

T ,U


Ts
Nu x 
hx x
k
1
 0.332 Re x 2 Pr
1
3
1
1
hx x
2
Nu x 
 0.664 Re x Pr 3
k
For mixed laminar and turbulent flow over flat plate
L
1  xc

hx    hlam dx   hturb dx 
L0

xc
45
Nu L   0.037 Re L  871 Pr1 3


0.6  Pr  60
5  10 5  Re L  108
Re x, c  5  10 5
Eq. 7.41

Heat Transfer
#5

Su Yongkang
School of Mechanical Engineering
External Convection Flow
For flow over cylinder
Overall Average Nusselt number
hD
1 3  Pr
Nu D 
 C Re m
Pr
 Pr
D
k
 s
14



Table 7.2 has constants C and m as f(Re)
For flow over sphere
 
hD
12
23
Nu D 
 2  (0.4 Re D  0.06 Re D ) Pr 0.4 
k
 s
14



For falling liquid drop
Nu D  2  0.6 Re1D2 Pr1 3
Heat Transfer
#6
Su Yongkang
School of Mechanical Engineering
Convection with Internal Flow
• Main difference is the constrained boundary layer
ro


• 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 )
Heat Transfer
#7
Su Yongkang
School of Mechanical Engineering
Internal Flow (Cont’d)
• For constant heat flux:
Ts (x)
T
Tm (x)
Tm, x 

qconv
 x  Tin
m c p
x
x fd ,thermal
• For constant wall temperature
if Ts  Ti
T
if Ts  Ti
T
Ts
Tm
Tm
Ts
x
• Sections 8.4 and 8.5 contain correlation
equations for Nusselt number
qconv  As h TLM
Heat Transfer
#8
x
Su Yongkang
School of Mechanical Engineering
Free (Natural) Convection
Unstable,
Bulk fluid motion
Stable,
No fluid motion
• Grashof number in natural convection is
analogous to the Reynolds number in forced
convection
GrL 
g  Ts  T  L3
2

Buoyancy forces
Viscous forces
GrL
GrL
Natural
Natural
 1 convection can
 1 convection
2
2
Re L
Re L
be neglected
Heat Transfer
dominates
#9
Su Yongkang
School of Mechanical Engineering
Free (Natural) Convection
Rayleigh number: For relative magnitude of
buoyancy and viscous forces
Rax  Grx  Pr
For vertical surface, transition to turbulence at Rax  109
• 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 9.309.32. (P513)
• Please refer to problem 9.34.
Heat Transfer
# 10
Su Yongkang
School of Mechanical Engineering
T A,out
Heat Exchangers
TB ,in (shell side)
Example:
TA,in (tube side)
Shell and Tube:
TB ,out
Cross-counter Flow
• Two basic methods discussed:
1. LMTD Method
q  UA
Tout  Tin
To
ln
Ti
 UATLMTD
2. -NTU Method
q   qmax
or :
q   C min Th,i  Tc ,i 
NTU 
Heat Transfer
 
where : qmax
q
qmax
 C min Th,i  Tc ,i 
  f  NTU , Cr 
Cmin
C r  1
Cr 
Cmax
UA overall , HX
C min
# 11
Su Yongkang
School of Mechanical Engineering
Discussion on the U
• Equation 11.5
Example 11.1
Notice!
1
1
1


UA U i Ai U o Ao
Rf,i ln( Do / Di ) Rf,o
1
1





hi Ai
Ai
2kL
Ao ho Ao
• For the unfinned, concentric, tubular heat
exchangers.
• When the inner tube surface area is the reference
calculating area.
Rf,o Ai
ln( Do / Di )
A
1
1
  Rf,i 
Ai 
 i
U i hi
2kL
Ao
ho Ao
• When the inner tube surface area is the reference
calculating area.
Rf,i Ao
ln( Do / Di )
Ao
1
1


  R f ,o 
Ao 

U o ho
2kL
Ai
hi Ai
Heat Transfer
# 12
Su Yongkang
School of Mechanical Engineering
Discussion on the problems
Heat Transfer
# 13
Su Yongkang
School of Mechanical Engineering