Convection Heat Transfer
Text Books：
1. Fundamentals of Heat and Mass Transfer：5th ed By Frank P Incropera and David
P. Dewitt John Wiley & Sons (Chap 6~10)
2. “Convection Heat Transfer” 3rd ed By Adrian Bejan;
John Wiley & Sons
Outlines：
1. Fundamental Concepts and principles
2. Forced Convection in Laminar Flows
(1) External Flow (Boundary Layer Theory)
(2) Internal Flow
3. Free Convection (or Natural Convection)
(1) External Flow
(2) Internal Flow
4. Convection with Change of Phase
1
Chap 1：Fundamental Concepts and Principles
§ 1.1 Introduction
§ 1.1.1 What and How?
Heat Transfer (or Heat) is energy in transit due to a temperature difference.

T
T
T
 T  i x  j y  k z


P
P
P
Gradient  P  i  j  k
x
y
z


C
C
C
 j
k
 C  i
x
y
z

 In a medium : T  0  q  0
△ 
 Be  tween two media : T  0  q  0
△ How? Three types of heat transfer process-Three modes.
1. Conduction：
Molecular diffusion of heat (by direction contact) heat transfer occurs across
the medium.
2. Convection：
(1) Random molecular motion (diffusion)
(2) Moving fluid (i.e. or macroscopic motion of the fluid)
3. Radiation：
Electromagnetic wave or photons (without working medium), All surfaces of
finite temperature will emit energy in the form of electromagnetic wave.
§ 1.2 Basic Laws
1. Fourier Law of Conduction
q w  k
T 
   kT
y  w
Where k ：Conductivity of media a material property
2
Typical Values of k w / mk 
Nonmetallic liquid
0.08 ~ 0.7
Alloys
14 ~ 20
Pure metals
50 ~400
2. Newton’s Law of Cooing
qw  h(Tw  T )
Where h( w / m2  k ) , called the convection heat transfer coefficient, the film
conduction is not a property.
It (h) depends on condition in the boundary layer, which are influenced by surface
geometry, the nature of the fluid motion, and an assortment of fluid
thermodynamics and transport properties.
Table.1.1 Typical values of the convection heat transfer coefficient process
h (w / m
2
 k)

Free Convection
Forced Convection
Gases
Liquids
2~25
50~1000
25~250
50~20,000
Convection with Phase
Change
Boiling or condensation
2500~100,000
Note：The fluid flow results from either an imposed pressure drop or an induced
buoyancy respectively called-forced and free convection.
3. Nu and Bi
T 

q w  h(Tw  T )   k f

y  w
C o n v e c t i o
n 

Co n d u c t i o n
 h  k f
T / y w  k f  (Tw  T )
 
(TW  T )  l  ( y / l )(Tw  T )
3
Nu 
hl T *
 *
kf
y
y* 
y
l
T* 
TW  T
TW  T
Where, Nusselt Number is the dimensionless fluid temperature gradient at the
surface (or wall)
But, the Biot Number
Bi 
Internal thermal resis tan ce of a s o l i d
hl

k(s)
B o u n d alrayy et rh e r m rael s i s t e n c e
§ 1.3 Transport Phenomena
Due to non-uniform distributed field
(1) Transport of momentum velocity gradient→momentum transfer
w  
u
y
viscous stress
(2) Transport of Heat
4
Temperature gradient→heat transfer
q w  k
T
y
or  kT 
(3) Transport of mass
Concentration gradient→mass flux
N  D
c
y
or  Dc
§ 1.4 Kinematics
(1) Mass Eq.
(2) Momentum Eq. (of motion)
(3) Energy Eq.
Conservation
1. Mass Conservation
Dmc.v
0
Dt
mcv     cv  dv 
Dm cv
D

 dv
Dt
Dt   c.v
Dm sys

     dv    V n dA
c .v
c. s
Dt
t
The mass instantaneously trapped inside the control volume.
mc.v

t

c. s
 V n dA  0  C.V 內部質量淨增率 + C.S 淨流出質量率 = 0

流經具有面積 A 的控制表面，質量流率 m ：
5

m   Q  A V
mc.v
 0 ；  eVe Ae   iVi Ai  0
t
2-D：differential element
當 Steady：
mc.v
  m i   m e
t
i

u
v
( dxdy)  ( udy  vdx)  ( u 
dx)dy  ( v 
dy )dx
t
x
y
  u v 
 t  x  y  dxdy  0


 u v 
 


u
v       0
t x
y
 x y 


 V      V  0


t
( v )

D 

 V  
Dt
t

V  ui  v j

   ( V )  0 ---------------------------------Continuity Eq.
t
  V  0 Steady flow
  V  0 in-compressible flow (   constant)
6
In cylindrical coordinate：
Vr Vr 1 V Vz
 

0
r
r r 
z
and spherical coordinate：
1 V
1  2
1 
0
( r Vr ) 
(V s i n ) +
sin  
r r
s i n 
2. Eq. of Motion（Momentum Eq.）
Analytical approach：differential control volume.
principle：2nd Law of Newtonian Motion

dV  dxdydz
dm  dV

DV
d F  (dm)
Dt

 V
d F S  d F B  ( dV )
 V  V
 t



7
Where
F S ：Surface force ； F B ：Body force
 V
V
V
v 
 d F S   gdV  ( dV )
u
v
w 
 t
x
y
z 

x-direction：
 yx
 x

dx   x )dydz  ( yx 
dy   yx )dxdz  ( zx  zx dz   zx )d x d y
x
y
z
dFSx  ( x 
 yx  zx 
 
d x d y d z
  x 


x

y

z


 x  yx  zx
Du


 g x  
x
y
z
Dt
Similarly
y-direction：
 y  xy  zy
Dv
g y 



y
x
z
Dt
z-direction：
g z 
 z  zy  xz
Dw


x
z
y
z
Dt
Newtonian Fluid：
 x   P    V  2 
2
3
u
x
 y   P    V  2
2
3
v
y
2
3
w
z
 z   P    V  2
 xy   yx
 yz   zy
 v u 
    
 x y 
 w v 
  
 
 y z 
(Ⅰ)
 u w 


 z x 
 zx   xz   
Ref：White F.M viscous Flow 2nd ed, N.Y McGraw-Hill 1991
8
Newtonian Fluid Eq.： （將 Eq.(Ⅰ)帶入 x, y, z-direction）

Du
P    u 2
    u v     w u 
 g x 
   2    V         
 
Dt
x x   x 3
 y   y x  z   x z 

    v w 
Dv
P    u v     v 2

 g y 
        2    V     
Dt
y x   y x  y   y 3
 z   z y 

Dw
P    w u     v w     w 2

    2
 g z 
  
     
   V 
Dt
z x   x z  y   z y  z   z 3

For incompressible flow, with constant viscosity.
N-S Eq. reduce to
In vector notion

DV
  g  P   2 V
Dt

V  ui  v j  wk
P  i
where,
P
P
P
j
k
x
y
z
2V  i
 2V
 2V
 2V

j

k
x 2
y 2
z 2
For steady flow：
DV V

 V  V
Dt
t
 (V  )V   g  P    2 V
Special case：
In-viscid fluid ：   0
Fluid in static： V  0 ；  g  P
2. Energy equation：（The First Law of Thermodynamics）
Recall：c.v
9
Qc.v  Wkc.v  mi hi  me he  d mec.v
e 
total
energy
u
int ermal
energy

v2

gz
2 位能

動能
 ：傳遞累積的量
d ：變化的量
For the control volume of finite size xy in the fig, the first Law of
Thermodynamics requires that：
{Ⅰ} = {Ⅱ} + {Ⅲ} + {Ⅳ} + {Ⅴ}
Where
{Ⅰ}：The rate of energy accumulation in the c.v
{Ⅱ}：The net transfer energy by fluid
{Ⅲ}：The net heat transfer by conduction
{Ⅳ}：The rate of internal heat generation ( i.e. electrical power dissipation )
{Ⅴ}：The net work from the c.v to its environment
此圖說了 {Ⅰ} 和 {Ⅱ} 部分
10
e (總能)  internal energy(內能) + kinetic energy(動能) + potential energy(位能)


{Ⅰ}  (xy )  u 
 t




{Ⅱ}  (xy ) ue   ve
y
 x

此圖說明 {Ⅲ} {Ⅳ} 和 {Ⅴ} 部分
 qx'' q 'y' 
 ;

{Ⅲ}  (xy )
y 
 x
'''
{Ⅳ}  q ( xy )
11
 u
v
u
v 
{Ⅴ}  (xy )  x
 y
  xy
  yx 
x
y
y
x 

  x
 y
 xy
 yx
 (xy ) u
v
u
v
 x
y
y
x





Note：The work transfer rate associated with normal stresses acting on the right side of the
c. v. is positive and equal to
 x
u 


x  u 
x xy
 x 
x
x 


The net work transfer due to Normal stress ( in x-div. )
 x 
 u
u
 x
xy
x
x 

Similarly, work transfer due to shear stress in x-dir.
 xy

u
   xy
xu
y
y


xy

Sub. into above {Ⅰ}、{Ⅱ}、{Ⅲ}、{Ⅳ} and {Ⅴ} into
 q x'' q 'y'  ''' 



e  ue  ve      q   xu     y v    xy u    yx v
t
x
y
y 
x
y
y
x
 x
 
{Ⅰ}
{Ⅱ}
{Ⅲ}
{Ⅳ}
 
 
{Ⅴ}
Consitutive relations：
Newtonian Fluid :   V
u 2  u v 
   
x 3  x y 
 v u 
    
 x y 
 x   P  2
 xy
D__   __ 

 V  ___ 
Dt
t
應變數
De   e 

 V  e 
EX：
Dt
t
12
De  e 
e 
e

u
v
Dt
t
x
y

De
 DP

 e
   V     q   q   P  V   ____________________(1)
Dt
 Dt

 u  2  v  2   u v  2
  2         
 x   y    y x 
1
enthalpy： h  e  P 

Dh De 1 DP P D



Dt
Dt  Dt  2 Dt
and Fourier Law of heat conduction
q ''  kT
Finally,

Dh
DP
P  DP

   kT   q ''' 
   
   V 
Dt
Dt
 Dt


0

Dh
DP
   kT   q ''' 
  ________________________________(2)
Dt
Dt
Thermodynamics property relations：
dh  TdS 
1

dP
dS 
S 
S 
 dT 
 dP
T  P
P T
CP 
h 
S 
 T
 dP
T  P
T  p
C
S 
  P
T  P
T
Maxwell relation：
 1   
S 
1 P 

  2
 
 
P T
T  P

 T  P
Where
 
1  
  Thermal expansion coefficient
 T  P
13
 dh  C P dT 

1

1  T dP
Dh
dT
DP
 C P
 1   T 
Dt
dt
Dt
-
____________________________________________(3)
Sub. Eq .(3) into Eq. (1) and Eq.(2)
Energy Eq. in terms of Temperature
DT
DP
   kT   q '''  T
  ________________________________(4)
Dt
Dt
  0 , for incompressible fluid,
C P
C P
DT
DP
   kT   q '''  1 
  _________________________________(5)
Dt
Dt
For ideal gas：
1 P 
1
 
 T  P T
P  RT  Pv  RT
 
For incompressible liquid,   0
DT
   kT   q '''   _______________________________________(6)
Dt
C P
For constant fluid conductivity (k), zero internal heat generation； negligible viscous
dissipation  , the energy yields：
DT
 k 2T ____________________________________________________(7)
Dt
C P
Steady flow, 2-D
  2T  2T 
 T
T 
  k  2  2 
v
y 
y 
 x
 x
C P  u
  2T  2T 
 T
T 
 u
    2  2 
v

x

y
y 


 x

k
 Thermal diffusivity coefficient
C P
Heat conduction （fluid in static）
  2T  2T 
T
   2  2 
t
y 
 x
P.S. 參考 Text Book[2]P352~P327
14
i. e.
C P
DT
 k 2T   ________________________________________________(8)
Dt
3-D case:
Cartesian coordinates (x, y, z)
  u  2  v  2  w  2 
  2T  2T  2T 
 T
T
T
T 




C P 
u
v
w
 k  2  2  2    2         
x
y
z 
y
z 
 t
 x
  x   y   z  
 u v  2  v w  2  w u  2  2  u v w  2 
  
  ____________(9)
      
     

 y x   z y   x z   3  x y z  
cylindrical coordinate (r,θ,z)
  v 
 1   T  1  2T  2T 
T v T
T 
 T
C P 
 vr

 vz
 2    2 r 
  k
r
 2
2
r
r 
z 
z 
 t
 r r  r  r 
  r 
2
v
v 
1  1 v z v 
1  v
1 vr   v z 
1  v
 1 v vr 

   

      
 
   r  z
r 
2  r 
z 
2  r
r r    z 
2  z
r 
 r 
2

2
2
2

 2 
1
  V   ________________________________________________________(10)
3

For steady-state, incompressible and constant-property, 2-D flow with negligible

dissipation.
  2T  2T 
 T
T 
  k  2  2 
C P  u
v

x

y
y 


 x
or
  2T  2T 
T
T
u
v
   2  2  ____________________________________________(11)
x
y
y 
 x
where  
k
C P
thermal diffusivity
Eq. of motion:
  2u  2u 
u
u
1 P
u
v

   2  2  _____________________________________(12)
x
y
 x
y 
 x
u
  2v  2v 
v
v
1 P
v  
   2  2  _____________________________________(13)
x
y
 y
y 
 x
15
2
continuity:
u v

 0 __________________________________________________(14)
x y
For two-dimensional flow along a curved wall
(法一)成大，陳朝光
u1 
e
R

ex
h1 R  y
u 2 
e
 ey
h2
R  y d  R  y  dx  dS1  h1dx
R
e x
1
  ey
x
R
e y
1
  ex
x
R

ex
 e y

e y

 e x

e x
0
y
e y
y
0
16
(法二)楊老師
d e y  1  d e x
d ey
dx
d ey
dx
 ex

d y

d
ex
R
e x
e x
d ex
1
 e y ；

  ey

x Rd
R
e x
0
y
e y
y
0
曲線座標：

ex  e y 
R




ex
 ey
hx x h y y R  y x
y
V  u e x  ve y
 V  0
 R

 

ex
 e y   u e x  ve y  0
y 
 R  y x





e y v
e
R u
R
R
e x ,e x 
uex  x 
ve x 

e y ,e y
R  y x
R y
x R  y
x y

17
R u
v
v


0
R  y x R  y y
Home Assignment #1
For two-dimensional flow along a curved wall shown in the following figure, please
use the following relation
e x
1
  ey
x
R
e y

1
  ex
x
R
ex
 e y

e y

 e x

e x
0
y
e y
y
0
To transform the following vector equation in terms of x-component and y-component
of the momentum equation: 
DV
 P   2 V
Dt
18
Chap2： Forced convection in Laminar Flow
§ 2.1 The concept of Boundary Layer
Boundary Layer theory was proposed by Prandtl shortly after the completion of
his doctoral dissertation in 1904.
The Velocity Boundary Layer
The Thermal Boundary Layer
§ 2.1.1 The Velocity B.L
 The quantity “δ”is termed the B.L thickness and it is typically defined as the
value of y for which u  0.99u
 With increasing distance from the leading edge, the effects of viscosity penetrate
farther into the free stream and the B.L. grows (i.e.   x ).
 The velocity B.L. is of extent  x  and is characterized by the presence of

 u

u  
 0  and shear stress  w    
velocity gradients 

y  y 0 
 y


 For external flows, it provides the basis for determining the local friction
coefficient
19
Cf 
w
1 2
u
2


u 

y 0
1 2
u
2
§ 2.1.1 The Thermal B.L
 δ(x), termed the B.L thickness is typically defined as the value of y for which
the ratio
TW  T
 0.99
TW  T
 With increasing distance from the leading edge, the effects of heat transfer
penetrate farther into the free stream and the thermal B.L. grows.
 At any distance x from the loading edge, the local Heat flux may be obtained by
applying Fourier Law to the fluid at y=0. That is
qw''  k f
T 

y  y  0
The above (relation) expression is appropriate because at the surface, there is no
fluid motion and energy transfer occurs only by conduction.
Further, with Newton’s Law of cooling
qw''  hTW  T 
20
kf
 h
TW
T 

y  y 0
 T 
TW  T  cons tan t
t  x

T
1 1


y  t x
 h and qw'' 
1
( h and qw'' decreases with increasing x )
x
 In summary the thermal B.L. is of extent  x  and is characterized by
temperature gradients and heat transfer.
§ 2.2 Similarity solution for Laminar Boundary-Layer-Flows
(A) Introduction
Uniform flow post a flat plate

y
 x 
Similarity Variable --------------------------------------(1)
For the same layer  , velocity is the same
21
u
 function of  ；  x, y   Stream function
u
 x  ~
x
Re x
x

u x

Re x 

y

u x x u  x



y
u
x
Similarity Analysis：
Find some variables to transform the P.D.E. to some O.D.E ’s
Ref：   xu f ( )
u


；v  
x
y
Continuity：
u v

0
x y
N-S Eq.：
u
u
 2u
v
  2 ------------------------------------------------------(2)
x
y
y
For steady, incompressible flow：
u
u
u
1 P
 2u
v

 g x  2
x
y
 x
y
u  Constant (external, region of B.L.)
u
v0
du
 1 P
u
P
Recall: d(    gy )  0
u  
2

dx
 x
O.D.E： ff " ( )  2 f " ' ( )  0
y  0；u  v  0
B.C：
2
y   ； u  u

  0； f  0 ； f '  0
  ； f ' 1
22
(B) steady-state ;
incompressible 2-D Laminar Boundary Layer pressure gradient
External flow (outside B.L)
u∞(x)
u
y
u
U
du
1 p

dx
ρ x
x
(Pressure gradient)是因為
有曲面的產生，若是平板
則沒有。
ß
Wedge ; 楔型
( Flow past a wedge )
u
du 
1 p

dx
ρ x
Continuity equation:
u v

 0 -----------------------------------------------------------------------------(1)
x y
Momentum equation:
u
du
u
u
 2u
v
 u     2 -------------------------------------------------------(2)
x
y
dx
y
The B.C’ s
(1) y=0 ; u=v=0
(2)y→∞ ;
(no-slip wall condition)-----------------------------------(3)
Lim u  u (x) ---------------------------------------------------------(4)

y 
at the entrance :
x=(x0)=0 , Let u=u2(y)------------------------------------------------------------------(5)
We introduce a stream function  (x, y) defined
u

y
; v

------------------------------------------------------------------(6)
x
and inserting (6) into (2) yields  (7)
Eq. (2) , yields
23
du
  2   2
 3
 u    3
y xy x y 2
dx
y
-----------------------------------------(7)
 ( x, y )   ( x ) f ( ) ----------------------------------------------------------(8a)
with
η=y*g(x)=function of x , y------------------------------------------------------------(9)

 ( x ) f ( )
Lim u  Lim
 Lim
y
y
y   y y  
--------------------------------------(10)

Lim  ( x )
f ' ( )  u ( x )

y
y
Considering Eq. (10)
f ' ( )
say
y 
 
 f ' ( y * g ( x )) y  cons tan t  1

 g ( x)
y
變成
Eq(10) 
Lim  ( x ) g ( x ) *1  u ( x )   ( x ) g ( x )  u


y
變成
u ( x)
Eq(8a )   ( x, y )   ( x ) f ( )  
f ( ) ----------------------------------(8b)
g ( x)
Substituting Eq(8b) into Eq(7) yields
u g'
g2
( f ' )2  (1   ) ff "  1  
f " ' --------------------------------------------(11)
u' g
u'


where
dg
dx
df
f '
d
g' 
du
dx
2
d f
f" 
d 2
u'  
f" ' 
d3 f
d 3
24
say
u g'

 const  b
u'  g

為使Eq(11)變成O.D.E所所作假設條件
2
say
g

(b)
 const  a

u '
，並藉此條件，聯立求 解u  ( x) g ( x)
(a) 1 -
au'  ( x )
 g ( x) 

--------------------------------------------------------------------(12)
g ' 1 u" 

g 2 u' 
Sub (12) into condition (a)
u g'
u 1 u"
 b
1 
1 
u' g
u' 2 u'



u"
u'
   2(1  b) 
u'
u


 u  (C1 x  C2 )
1
2 b 1
d ln u'   2(1  b)d ln u
ln u'   2(1  b) ln u
u'   c'1 u
u
22 b
22 b
du  c'1 dx
1
2 b 1
u
 c1 x  c2
2b  1
C1; C2  arbitary constant
case(1) b≠1/2
u ( x )  (c1 x  c2 ) m -------------------------------------------------------------------(13a)
where m 
case(2)
1
2b  1
; b
m 1
2m
b=1/2
u ( x )  c3e c4 x -------------------------------------------------------------------------(13b)
c3 ; c4 arbitrary const
Let a=b (arbitrary be chosen at will)
Eq. (11) yields
f " ' ff "  1  ( f ' ) 2  0 ------------------------------------------------------------(14)


25
 
1
2m

b
m 1
Falkner-Skan Equation
case (1) b≠1/2
au'
  y m  1 c1 (c x  c ) m1
1
2

2 
-------------------------------(15a)
m  1 c1 ( m1) / m
y
u
2 
  yg ( x )  y
m 1
2 
u m * f ( ) ---------------------------------------(15b)
m 1 c 
1
u
   f ( ) 
g
u( 

)  u * f ' ( ) ---------------------------------------------------------------(15c)
y
v(  
m 1

m 1
m 1
)
c1u m ( f 
f ' ) ----------------------------------(15d)
x
2
m 1
B.C (3) (4)
 f (0)  0
f ()  1
case(2) b=1/2
y

----------------------------------------------------------(16)
; β=2
c4 u 
; u  u  f ' ( )]
2
---------------------------------------------(17)
2u
c4 u
f ; v( f  f ' )
c4
2
Wedge-Type flow :
u∞(x)
y
x
U
ßπ
26
Potential (inviscid) flow
Theory shows the free stream velocity at the wedge surface varies with distance from
the tip as u∞(x)=cxm ; where m=ß/2- ß
Case(1) c1m=c ; c2=0 sub. Into (13a)=> u∞(x)=cxm
(c1,c2 為任意常數);
y
m 1 c
( cx m )
2 
Case(2) at
m 1
m
y
x=0 ß=2
（由 13.a 式）
m  1 c m 1
x ----------------------------------------(18)
2 
b=1/2
u∞=c3e0=c3≠0
b=1/2在 flat plate u 
du 
 0(才可用 )
dx
η=y
當 ß=0 (m=0)
Flow over a flat plate
u
1 u
y
~
 y  --------------------------------------------------------(19)
2 x  ( x )
x
similarity variable
y
solution of Eq(14) & B.C’ s (16)
can be obtained F.D.M (Runge-Kutta)
u
f 
depends on η ; but independent of x (similarity)
u
The shear Tw at the wall is particular engineering interest.
 u  u x * f  
w  
u
y
y 0   * u  *
d2 f
d 2
0
*

y
0

u 
m 1
* u  *
…………（20）
2
x
27
補充觀念：[1]
p
1 2
  const
 2 
u
(
 0) potential theory flow  適用白努力方程式
y

u
 0
y
B.C
適合 potential flow 是
不考慮黏滯性
Ideal fluid (   0)(   const)

 V  0

V  
 velocity potantial





v
  (V  )V  p  g
t
補充觀念：[2]
(c) steady laminar forced convection in incompressible wedge flow with
non-uniform wall temperature
y
T∞
u∞(x)
u
U
TW(x)
B.L
x
ßπ
Assuming constant properties, the energy equation is
2
T
T
 2T   u 
u
v
 k 2    ----------------------------------------------------(21)
x
y
y
c p  y 
with B. C’s
28
(1) y=0
T=Tw(x)-------------------------------------------------------------------(22)
(2) y→∞
T=T∞----------------------------------------------------------------------(23)
and the entrance condition is x=0 T= T∞------------------------------------------(24)
For wedge Flow u∞(x)=cxm-----------------------------------------------------------(25)
Where m=ß/2- ß
And ßπ is the inclined angle of the wedge
By similarity solution , we may assume
u  u f ' ( ) ---------------------------------------------------------------------------(26a)
v  u 
m 1  
m 1 
f 
f  ----------------------------------------------(26b)
m 1 
2 cx 
m 1 
similarity temperature field
T  T  X ( x) ( ) ------------------------------------------------------------------(27)
where
y
m  1 c m1
x  ( similarity variable )
2 
Comparing Eq. (22) and (27) yields
T  Tw ( x ) (  0 ;  (0)  1)
Tw (x)  T  X(x) --------------------------------------------------------------------(28)
y→∞
η→∞ X(x) →0
From Eq.(23) & (24) ;  ( )  0  
T  T
Tw ( x )  T
Using Eq.(18) , (26) & (27) ; Eq(21) can be obtained as follow
 " Pr f '
2 xX '
u 2  Pr
Pr f ' 
( f " ) 2  0 -------------------------------(29)
m 1 X
X ( x) C p
非線性混合 O.D.E
To make Eq(29) to become O.D.E function of η alone, we require the following
conditions.
29
xX'
indep. of x  n  X(x)  Ax n
---------------------------------------------(30)
X
A  int egral const.
(a )
u 2
(Cx m ) 2
indep. of x 
to be const.
X
Ax n
-----------------------------------------(31)
n
n  2m m 
2
(b)
In the presence of dissipation
Tw ( x )  T  AX 2 m m 

2-
(29) simplifies to  " p r f '
------------------------------------------------------(32)
c p
4m
Pr f '  2 r ( f " ) 2  0 -------------------(33)
m 1
A Cp
with B. C’s
(1)   0  (0)  1 -------------------------------------------------------------------(34a)
(1)     ( )  1 ----------------------------------------------------------------(34b)
If dissipation is negligible
Tw ( x )  T  Ax n ----------------------------------------------------------------------(35)
 " Pr f '
2n
Pr f '  0 ------------------------------------------------------(36)
m 1
(Note：n=2m 可不一定成立，2/m+1=2-ß)
special case : X(x)=Axn=> X(x)=A
(a ) n  0  Tw ( x)  T  A  const (constant wall temperatu re) （Isothermal wall）
(b)   0   " Pr f '2 Pr f '  0 (flow over flat plate) ---------------------(36’)
Ref : f”’+ff”=0-----------------------------------------------------------------------(Eq14)
θ and f’ and equivalent
Runge-Kutta method : (數值分析法) Pr 
0.01
0.1
0.7
1

k

;   ; 

C P

10
102
103
Pr
Liquid metal
Water
fig.（1）
Oil
30
(Pr  1 ; velocity B.L. is similar to thermal B.L)
metal t
Pr  1 (Liquid metal)   1 ; t  
 

t
Pr  1 (Oil)   1 ; t  

θ
1
0.8
0.6
Pr=0.5
0.4
1
0.2
50
3
7
15
1
2
η
Fig (2)
Temperature profile for Laminar B.C Flow over a flat plate with constant wall
temperature ( i.e. θ”+Pr f θ’=0)
u∞
金屬液體
δ
δt
L
金屬液體δt >一般的液體δ
k
h
T
y
k
0
Tw  T
~
T
t
T
Pr=0.7
0
~
k
t
hL
Nu 
k
θ
1
0.8
n=-0.5
0.6
0
Pr=0.5
0.4
0.5
1
0.2
4.0
3.0
3
7
15
1
2
3
η
4
Fig (3)
Temperature profile for Laminar B-L Flow over a flat plate with variable wall
temperature.
The heat flux qw from the wall into the fluid is
qw   k
T
y
  k (Tw  T )
w ( y 0 )( 0 )
d
d
0
m  1 u
------------------------(37)
2
x
31
qw  h(Tw  T ) ----------------------------------------------------------------------(38)
 37  38
Note:   0 ;
d
can be evaluated form Fig(1) and (3)
d 0
h  k
m  1 u  d
-------------------------------------------------------------(39)
2
x d  0
Nu ( x) 
hx
m  1 d

k
2 d
Re x -------------------------------------------------(40)
0
( n=0 ; n≠0 may apply)
 ' (0)  0.332 Pr1/ 3 (if Pr  0.5)
Nu  0.332 Pr1/ 3 Rex1/ 2 ; Pr  0.5
Temperature profiles for Laminar wedge-type flow with constant surface
Temperature.
θ
1
0.8
0.6
0.4
ß=1
0.2 ß=1.6
Pr=0.7
ß=1
ß=0.4
ß=0.4 ß=1.6
Pr=1.0
1
2
3
4
η
Fig (4)
3
u∞
ßπ
Pr=10
Pr=5
2
 
2m
m 1
u  cx
m
1
Nu
N up
0 0.2

2
Pr=0
ß=1.6
2
0.6 0.8 1.0 1.2 1.4 1.6
Ration of Nu for Laminar wedge-type flow to Nu for flat-plate flow at constant
surface temperature h=0.
32
Nu/Niso
Tw-T∞=Ax2
non-uniform/
4
m=0
m=1/2
3
m=1
等溫平板
2
Turbulent
1
2
4
6
8
10
Xn
Fig (5)
Ratio of Nu for surface with temperature profile locally varying temperature to the Nu
for an isothermal surface , wedge-type flow with Laminar and turbulent B-L
Low Pr
Nu
1
Nux=0.332(Rex)1/2Pr
Nux=0.5(Rex Pr)1/2
1/3
(Exact similarity sol.)
High Pr
0.1
ß=0
0.01
10-3
10-2
10-1
1
10
n=0
Pr
Fig (6)
(Fig 6)
Nusselt no. for Laminar B.L. flow of fluid with various Pr over a flat plate
Nu  Pr 1/ 2 for Pr ~ 0.01 ( 0)
Nu  Pr 1/ 3 for Pr ~ 102 ( )
For constant wall temperature n=0 (36) yields
 " Pr f '  0 --------------------------------------------------------------------------(41)
33
d '
 Pr fd  0
'
 '  C1e  Pr  fd


0
0
  C1  exp(  Pr  fd )  C 2
B.C
  0 ;   1  C2  1
  0 ;   0  C1 
1


0
0
 exp(  Pr  fd )d
 1- 


0


0
0
 exp(  Pr  fd )d
0
....................................................................................42 
 exp(  Pr  fd )d
 ' (0) 
d
d

0
1


0
0
..........................................................................43
 exp(  Pr  fd )d
(1) Pr  1 ;  t  
assume

d
d
Nu 
u
 1  f ' ( )  f  
u

0
1
2 1/ 2

Pr

Pr 2
0 exp(  2  )d
Thermal B.L.

Velocity B.L.
1 2 1/ 2
1
1/ 2
1/ 2
Pr Re x 
Pr 1 / 2 Re x
2 

 0.564 Re x Pr
(2) Pr  1 ;  t  
.
u
 c  f ' ( )
u
c
f ( )   2
2
d
1

 k Pr 1 / 3
3
d 0
c
0 exp(  Pr 6 )d
k  0.332 2
34
With these limiting values in mind , one finds that the Nu for flow over a flat plate
(ß=0) with constant wall temperature (n=0) can be well approximated by the
following :
Nu  0.332 Re x 3 Pr
Nu  0.5 Re x Pr
Tw ( x) ~ x n
for
for
Pr  0.6 -----------------------------------------(46)
0.005  Pr  0.05 ---------------------------------(47)
(variable wall temperatu re)
Tw  arbitary function of x
such a temperature variation can be expressed by the series

Tw  T   ai x i ----------------------------------------------------------------------(48)
i 1
If (33) diff. eq. is linear , we may apply superposition method

qw   k 
i 1
Ti
y
0
where Ti is the temperature distribution for (Tw-T∞)~xi with Eqs. (18) (26) (28a) and
(40) are obtained.
qw 

kN u 0
N
N
( a 0  a1 x u1  a 2 x 2 u 2  ......)
x
N u0
N u0
kN u 0
x

N ui
a x ( N
i
i 1
i
)
u0
n  0 const. wall temperatu re N u0  N uis Fig 5
where
Nui : The Nusselt no. for a wall-temperature variation according to Eq(35) with n=i
Nu0 : The Nusselt no. for a constant wall-temperature
35
Note :

Eq ( 48)
qw  h(Tw  T)  h  ai x i
i 1

kN u 0
N
N
( a0  a1 x u1  a 2 x 2 u 2  ......)
x
N u0
N u0

(T  T )
(Tw  T )
qw   k (Tw  T )
d
d
0
m  1 u
2
x
m  1 Cx m 1
2

 h(Tw  T )  h( a0  a1 x  a 2 x 2  ......)
 k (Tw  T ) ' (0)
qw x hx

( a0  a1 x  a 2 x 2  ......)
k
k
 a0 N u 0  a1 xN u1  a 2 x 2 N u 2  ......
Approximate Laminar B-L. Solution Employing the Integral Eq. and
Superposition
(A) Laminar B-L. over a flat plate integral approach
Consider the definition
u
 ( w)  
y 0 …... (1)
y
T
y 0
u
y
h
……… (2) Note
T0  T
y
k
And
0
is function of x only
 We have the apportioning to simplify the B-L equation by eliminating y variables,
this accomplished by integrating each equation term by term from y=0 to y=Y;
Where y  max{ ;  t } is situated in the external flow
Y
y
B.L
Y
 or  t
y=0
x
u0=0; no slip
v0=0; no blowing suction
36
Steady & incompressible, flow
u v

 0 ………………………… (a)
x y
Mass Eq
Momentum Eq
Energy Eq
u
u
u
v
1 p
 2u
v

  2 ……… (b)
x
y
 x
y
T
T
 2T
v
  2 …………………(c)
x
y
y
Substituting (a) into (b), (c)
u 2 v
1 p
 2u
 (uv )  
  2 ……… (3)
x y
 x
y
T
T
 2T
(uT ) 
(vT )   2 ………… (4)
x
y
y
Integrating (3) and (4) from y=0 to y=Y yields
d Y 2
1 dp
u
u dy  uY vY  u0 v0  


dx 0
 dx
y
d Y
T
u T dy  vY TY  v0T0  

dx 0
y
Y


Y
T
y
Assume the external flow (free stream) is uniform
0
u
y
0
……… (5)
……………… (6)


 y

  0 ; and

u  u
T  T ; and v0 .we evaluate vY by integrating the continuity equation.
d Y
u dy  vY  v0  0
dx 0
vY  
d Y
u dy ……………………………(7)
dx 0
Sub (7) into (5) and (6) yields
d Y
1 dp
u
u(u  u)dy  Y


0
dx
 dx
y
d Y
T
u(T  T )dy  

0
dx
y
0
0
………(8)
…………………(9)
Note: another approach: employing control volume energy balance, momentum
balance, to derive Eq. (8) (9)
37
u
y
C..

u( y)
x
W
u
u
u 2
C..
u
1
u 2
2

 W ,x
momentumu 1u Momentum balance
The conservation of mass for this C.. give

x
0
0
u    udy    vdy ……………………(*)
The momentum balance for this C.. yields

x
x
0
0
0
u 2    u 2 dy  u  vdy    w dx
Or, in terms of (*)

x
0
0
  u(u  u)dy    w dx ……………………(
Differential (
)
) with respect to x
d 
u
u(u  u)dy  

0
dx
y
0
Assume the velocity profile to be linear within B-L
u
y
y
  
u 

38
d  2  1 u
u 0
dx 
u
  y  d 
2
d   
 u 
    dx 

u
1 
 u
 u 
  
1
 u 
 
6
 y
 
 
y 0
u


u

Assume the velocity profile to be parabolic and cubic within B-L cubic
Parabolic
x

x
Cubic

 0.365 Re x
1
 0.323 Re x
1
2
2
Assume a velocity profile as
u  u g   ;0    1 ……… (10a)
u  u
;  1………… (10b)
Where  
y

dp
 0 (flat plate).
dx
d  1
 dg

g  1  g  d  
……… (11)

 u  d  0
dx  0
Sub. Eqs. (10a) (10b) into (8) with
The result expressions for local B-L thickness and skin friction coefficient are
x

 a1 Re x
C fx 
1
2
…………………… (12)
0
 a2 Re x
1
u 2
2
1
2
……… (13)
1
  dg   2
 
 2
d


0 
With a1   1
…………………………… (14)
 g 1  g d 
 0



1
  dg 
 2
1
  g 1  g d  ………………… (15)
a 2  2
  d  0 0

39
Table 1a: the impact of profile shape on the integral solution to the laminar B-L
friction.
g  
a1  Re x
Linear 
Cubic

3   
2
2
  
Sine sin 

 2 
Similarity sol.
Exact sol
1
a2  C fx Re x
2
1
2
Nu 
C fx
2
Re x
3.46
0.577
0.289
4.64
0.646
0.323
4.8
0.654
0.327
4.92
0.664
0.332
1
2
Pr
1
3
Note: Reynolds-Colburn Analogy
St Pr
2
3

C fx
2
; St x 
Nu x
Nu x

Re Pr Pex
1

 Pr 3 Pr  1;
t
Next, find temperature profile assume+
T0  T  T0  T g   ; 0   t  1 ……… (16a)
T  T ; g t   1 ;  t  1 ………………………… (16b)
For high Pr, fluid  t  
2
dg
d t
1
t 0
 1  t 

Pr 
g  t 1  g  t d t  ……… (17)
2 0

 t     
 a1 
  

Assume that the temp. Profile g   t (linear)
t
1
 Pr 3 ;

Nu  0.289 Pr
Assume g t  
t
2
3   
2
t
1
3
1
Re x 2 ……………………… (18)
(17) yields
40
t
1
 0.977 Pr 3 …………………………… (19)

1
1
k  x k Pr 3 Re x 2
h


………… (20)
 t x  t  x 0.977 4.64
k
Nu x 
1
1
hx
 0.323 Pr 3 Re x 2 ……………… (21)
k
Table 1b: The impact of profile shape on the integral solution to the laminar B-L heat
transfer problem
g  t 
Uniform surface
Uniform wall heat
temperature
flux
Linear  t
Cubic
3   
2
t
2
t
  
Sine sin  t 
 2 
Similarity sol.
Exact sol
Pr<<1;  t  
0.289
0.364
0.323
0.417
0.337
0.424
0.332
0.458
(Liquid metals cases)
Instead of (17), we obtain
dg
2

 0

1
d t t     t 
  
t g 
Pr 
1  g  t d t   g  t t 1  g  t d t  ……… (22)
t
2 0
 

  
t 
  t  
 a1 
  
The sums of two integrals stems from the fact that  t  
 u  u f ; for 0  y  
u  u  ; for   y   t
Assume g   t (linear) (22) yields

t
1
 3 Pr  2 ; Pr  1 ……… (23)

In other words (
t
x

 t  3.46  12
1

Pr Re x 2 )
 x
3
41
t
x
 2 Pr
1
2
Re x
1
; Pr  1 ……… (24)
2
By definition
h
k
t

1
k x
k 1
 2 Pr 2 Re x 2 ; Pr  1 ……… (25)
x t
x
Nu 
1
hx 1 12
 Pr Re x 2 ; Pr  1
k
2
補充
  t    t   t 3 (Pr  1)
3
2
1
2
d t

u  (1   )dy  

dx 0
y
Where  
T  T0
T  T0
; 1- 
T  T
y 0
   t  u  u 
T  T0
d t
3
1 3
3
u  (1   t   t )dy  

0
dx
2
2
2 t
 u
 h x  
Nu 
3 d t 3

8 dx 2 t
3k
3k

2 t 2 8
 t 
u
3 k

x 2 8 x
8x
u
u x 


1
1
hx
3

Re x Pr  0.532 Pr 2 Re x 2
k
2 8
Exact solution for Pr  0 ; Nu x  0.564 Pr
1
2
Re x
1
2
u
t

42
Integral method for laminar flow over flat plate with uniform heat flux ( q0  const ).
Table 1b
Nu x  0.417 Pr
1
2
Re x
1
3
Assume u  a 0  a1 y  a 2 y 2  a3 y 3
Boundary condition
1. y  0 ; u  0
2.
3.
4.
y  0 ; y   ; u  u
u
0
y
y0 ;
 2u
0
y 2
u
3 y 1 y

  
u 2  2   
3
 3 y 1  y 3 

u   u  u  1 

 2  2    


The integral momentum eqn.
u
u
2
2
d  u
dx 0 u 
:

u
1 
 u

u
dy  
y

0
 u 
 
u
u
   

 y
 
 
y 0

3 u 
2 
3
3
 3  y 
d
1  y   3  y  1  y  
3 u 
         1        dy 
dx 0  2    2      2    2    
2 
u
2
d  39  3 u 


dx  280  2 

d 140 

dx
13 u 
x  0 ;   0  2 
280 x
13 u 
43

x

280
13
1
Re x
0
C fx 
 0 3 u 

 2 
 4.64 Re x
1
u  2
2
13
280
 C fx  3

3 
 u
1

2
0.646
Re x
The integral energy equation
q
 T
d Y
Where q 0  k
u (T  T )dy  0

0
dx
C P
y
1
(Reference table 1a)
Re x

0
k
C P
(ref. 9)
Assume T  b0  b1 y  b2 y 2  b3 y 3
Boundary condition
 T
y
; or y   ; T  T
1.
y  0 ; q 0  k
2.
y  t
3.
 T
 2T

y0 ;

0
y 2
 y
4.
y   t ; or y  
0

 const q 0 

T
y
 t ()
0
2 q 0  3 y 1  y
 T  T 
t 1
 
3 k  2  t 2   t




3




For special wall flux heating, by setting y=0 the interrelation between the unknown
T0 and  t is found to be
T0  T 
2 q0 
T
 t  q0  k
3 k 
y
T  T0
3 y 1 y


 
T  T0 2  t 2   t



0



3
44
For the case: Pr 


1  t     t 1


3
3
q
d  t   3  y  1  y    2 q 0  t  3  y  1  y   
1        dy  0
u         


dx 0   2    2      3 k  2   t  2   t   
C P



 y   t ; T  T  uT - T   0

3
3
d  t  3  y  1  y    2 q 0  t  3  y  1  y     y
1         t d 
u           
dx 0
 2   t  2   t    3 k  2   t  2   t      t


q
  0
 C P
q
d  2 q0 u   1
1 3 
t
   0
 

dx 
k  10
140  C P
Note that x=0;  t =0;  
t
0

Integral
q0 u  2  1 3
1 5
1 x
 
q 0 dx
  
k
140  C P 0
 10
1 3
1 5
 1 x
13
1 x
 
 
q
dx

q 0 dx
0
10
140
280 Pr x q 0 0
u   2 q 0 0
1  
13 1 1
 t 

   t 
28 Pr x q 0
   14   
3
5

x
0
q 0 dx
Special case
q0  0
q 0  const
for x  x 0  0
for x  x 0  0
x
1  
13 1 q 0
 t 
x  x0   13 1 1  0 

   t 
28 Pr x q 0 x
28 Pr 
x 
   14   
3
5

1  
 
 t  1 ;   t    t 

14   
  
5
 13 1  x 0
t   
1 
x
 28 Pr 
x 
2 q0
2 q 0  13 1 
T0  T 
t 

1  0  
3 k
3 k  28 Pr 
x 
1
3



1
3
3
q
 2.4 0
k
x  1 
x0
 1 
u   Pr 
x



1
3
45
1
q0
x 0.417 Pr 3 Re x
Nu x 

1
T0  T k
x0  3

1  
x 

for x 0 =0
Nu x  0.417 Pr
1
2
Re x
1
2
1
3
(B) Unheated Starting
Assuming the temp. profile
T
u
Y
t

T0
T

T  T0

2
 g ( t )  t 3   t
T  T0
2
u

 g ( )  3   2
u
2

From (9)

Y
0
d Y
T
u (T  T )dy  

0
dx
y


t 

y
t
y

0
Y
Y3
1 
3
1 3
u (T  T )dy   g  1  g  d  u  T  T0      3 1   t   t dy
0
0
2 
2
2
2

let  
t

Pr  1 ,  
t

1 ;  

t
Y
d  t
T
u
(
T

T
)
dy

u (T  T )dy   





0

t

dx 
y
u
d t
dx

1 3

 3
1   t   t d t 
2
y
 2

d  3
3 3 
t t   
 
dx  20
280  u 
3
1
1
  2   2 
0
0
3
0
d  3 2
3 4 3 
………(*)
 
  
dx  20
280  2 u 
46
By momentum integral equation, we have
d  2  u 
u
1 
u  0
dx 
u  u
  4.64
x
u
  0
u
dy  

y
  

3 u
2 
1
 4.64 x Re x 2 ………(**)
Sub. Equation. (**) into (*) and neglecting
3 4
 term
280
d 13 1

dx 14 Pr
3  42 x
3 
0
4 x d3 13 1

3 dx 14 Pr
Boundary condition
x  x0 ;   0
3
13 1
 
 Cx 4
14 Pr
3
 13 1 


 14 Pr 
qx 
1
3
  x  34 
1   0  
  x  


3
3
1   x0  4 
 0.9756 1 1    
Pr 3   x  
1
3
3k T0  T  3 k T0  T 

 hT0  T 
2 t
2

3k 1 3
k
h

2   2 x 4.64 Re 21
x
1
hx 3 1
1
Nu x 

Pr 3
k 2 4.64 0.975
If we keep
1
3
13 x 0 4
C
14 Pr
3


1  13   x 0  4  


Pr
1

  

0.9756 
  x  



  x  34 
1
2
Re x 1   0  
  x  


1
3
 0.33136 Pr
1
3
1
3
  x  34 
1
2
Re x 1   0  
  x  


3 4
 ;
280

47
1
3
 2 
 1  
 14 
1
3
3
1   x0  4 
 0.9756 1 1    
Pr 3   x  
1
  x  34 
Nu x  0.33141   0  
  x  


3
1
3
 2 
1  
 14 
1
3
Pr
1
3
Re x
1
2
For Pr  1    1 if x 0 =0
1

Nu x  0.33141  
 14 
1
3
  x  34 
1   0  
  x  


  x  34 
 0.3231   0  
  x  


1
1
3
Pr
1
3
Re x
3
Pr
1
3
Re x
1
2
;
1
2
T  T0
T  T0
Chapter 2.3 the Circular Tube with Fully Developed Velocity and Temperature
Profile
Inviscous flow region
Boundary layer
u
dr
u
 r  d r
r
r0
r
r

Boundary layer region
Parabolic velocity profile
Fully developed velocity region
Hydrodynamic
(A) Fully Developed Laminar Flow In A Circular Tube
Fully developed:
u
0
x
Momentum equation:
u ( x, r )  u ( r )
F
X
v vr   0
radial velocity
0

p 



p 2rdr   p 
dx 2rdr   r 2rdx   r  r dr 2 r  dr dx  0
x 
r



48

d
r r   r dp
dr
dx
 r   
du
du
 xy   
dr
dy
y  r0 - r
   u 
dp
r  
r r  r  dx
 d  du  dp
…………… (1)
r

r dr  dr  dx
Boundary condition
r 0 ;
r  r0
u
 0 ……………… (2)
r
; u  0 ……………… (3)
Integrating directly yields
2
r0  dp 
r 2 


1

……… (4)


2
4  dx 
r0 
u
The total mass flow rate
m   u r dA  VdA ………… (5)
V
1
udA
A 
A  r0 ……… (6)
2
1
udA dA  2rdr ……… (7)
A
2 r0
(7)  V  2  u  rdr ……… (8)
r0 0
For incompressible flow V 
Substituting (4) into (8)
2
r  dp 
V  0  -  ………………… (9)
8  dx 
Combining (4) & (9)

r2 
u  2V 1  2  …………… (10)

r0 

The skin friction of coefficient
C fx 
0
1
V 2
2

u
r
V 2

r  r0

2
16
2r0 V


16
………… (11)
Re
Where
Re 
V (2r0 ) VD

…………… (12)


49
(B) Fully Developed Temperature Profile
r
r0
t
Fully developed region
Thermal entrance
The term fully developed temperature profile implies that there exists under certain
condition a generalized temp. Profile is invariant with tube length
  T0  T

x  T0  Tm

0


Function of r
Where T0 = tube wall temp and Tm = mean temp. In tube
Axial conserved thermal energy rate: m C p Tm
 C p Tm  VAC P Tm   uC P TdA
m
A
 Tm 
If
T0  T
T0  Tm
 uTdA
VA
………… (21)
is invariant w. r. t. x
We can write the condition at the wall
T

r  r0
   T0  T 

……………… (22)
 const  x
 
T0  Tm
 r  T0  Tm  r r0
q 0  hT0  Tm  ………… (23)
Note: for external flow q 0  hT0  T 
50
q 0  k
From (23) divided by (24) equals
k  c o n tan
s t  Thus
T
r
r  r0
………… (24)
h
 cons tan t
k
h  c o n tsa nt
 T T 

 0

 T0  Tm 
0
x
T dT0 T0  T dT0 T0  T dTm
…………..… (25)



x
dx T0  Tm dx T0  Tm dx
Assumption:
i. Steady laminar flow
ii. Low velocity
iii. Constant physical properties
iv. Vr  0
;
u
0
r
v. axial symmetric
vi. neglecting axial conduction
Energy Eq:
C P u
 1   T   2T 
T
T
 C vVr
 k
r
  2  …………… (26)
x
r
 r r  r  x 
From (IV) (VI)
 1   T 
T
 k
r
 ……… (27)
x
 r r  r 
1   T  u T

……………… (28)
r

r r  r   x
 C P u
Case (i) for the case of constant heat flux (per unit tube length)
q 0  hT0  Tm  =constant
h  cons tan t (For fully developed flow in tube)
 T0  Tm  cons tan t
51
d T0  Tm 
0
dx

dT0 dTm

dx
dx
Sub. Into (25) yields
T dT0 dTm
…………… (29)


x
dx
dx
Case (ii) next, for constant surface temperature
T0 =constant case
dT0
T T0  T dTm
0 

dx
x T0  Tm dx
For these two conditions (29) and (30) can be substitute into (28) to yield the two
differential equation.
Case1: 29
1   T  u dTm
……………… (31) For q 0  const
r

r r  r   dx
Case2: 30
1   T  u T0  T dTm
……… (32) For T0  const
r

r r  r   T0  Tm dx
Considering an energy balance on a control volume as shown in the following fig, we
have
q0 2x0 dx 

CPTm r02V



CP  Tm 

Tm  2
dx  r0 V
x


dx
 q0 
r0VC P dTm
………… (33)
2
dx
Consider
u T 1   T  d 2T

r

 x r r  r  dx 2
Scale analysis
D  2r0
52
q0
kD
:
T
D2

Pe >> 1
q0
1
D DC PU
:
q0 D
: 1
T k
:
=
hD
:
k
D hD 
L k UD
1
1
:
:
 T 
 2
D 
1
D q0 D 
L T k UD
=
= Nu :
For Nu  1
1
D
1
Nu
L
Pe
;D<L
 2T
0
x 2
axial conduction effect neglected.
(C) Constant-Heat-Flux Case
1   T  u dTm
 const  …………………(41)
r

r r  r   dx
Boundary condition
r  r0 ; T  T0
r 0 ;
T
0
r
From equation (10)

r2 
u  2V 1  2 

r0 


r2 
2V 1  2 

r0  dTm
1   T 


r

r r  r 

dx
 T  T0 
2V dTm  3 2
r4
r 2 
r0 

……… (42)
2
 dx  16
4 
16r0
From (21)
Tm 
 uTdA  T
VA
0

11 2V dTm 2
r0 ……………… (43)
96  dx
53
q 0  hT0  Tm   h
11 V dTm 2
r0 ………………… (44)
48  dx
Eq. (33)
r0VC P dTm
………………………………… (45)
2
dx
48 k
48 k

From (44) divided by (45), we have h 
……………… (46)
11 2r0 11 D
q0 
Nu 
hD 48

…………………………………… (47)
k
11
(D) Constant Surface Temp Case
1  T  u T0  T  dTm 

............................................................(32)
r

r  r   T0  Tm  dx 
Bc' s
r  r0
;
T  T0  c o n s t
r 0
;
T
0
r
Sub. Eq. (10) into Eq. (32) gives
1   T  2V 
r2
r

1



r r  r    r0 2
 T0  T  dTm

 T  T  dx
m
 0

..............................(51)

Successive approximate solutions is
N u  3.658
※ Exercise 將(42)式與(44)式帶入(51)式先求出 1st approx(51)式積分求得 Tm (x)
如此反覆直到 approaches a limit
(E) Find Tm (x) mixed mean fluid Temp
(i) From (29),(33)and(37)
dT0 dTm T
2q 0



 cons tan t...........q0  cons tan t
dx
dx
x r0VC p
T0
54
Tm
x
Fully develop
(ii) For (33) ; constant T0
q0  hT0  Tm ( x) 
A s s u m e xe  x1
;
dT
V
r0 C p m
2
dx
Tm  T1
 N u  x  x1  
...............(62)
 T0  Tm  T0  T1  exp  
2

r
V
0


T0
Tm
x
Thermally developed pipe and duct flows are characterized by a constant heat transfer
coefficient which is independent of x. There are four types of thermal. Bc' s that are
compatible with the existence of a thermally developed region
(i) T0  cons tan t (ii) q0  cons tan t (iii) q 0 ~ e x
(v) convective heat transfer from the external surface of the duct to a fluid
environment with he and T both of which are uniform.
2-4 Circular Tube Thermal Entry Length
(Velocity profile fully developed)
(Thermally developing laminar flow)
55
D
T
xtid
(A) Solution for uniform wall temp Dimensionless :
x
r
x 
; r 
r0
r0
T0  T
 T : Uniform in let temp
T0  T
Recall Chapter 3 Eq. (28)
t

1
  t  2r0 t
...........................(1)
 r

2
 x
r  1  r  r   r  


x  0
;
Bc r   0
;
r  1
;
t  1............(2)
t
 0........(3)
r 
t  0...........(4)
Try product sol , t  ( x ) (r )
Sub into Eq.(1) yields
1
1 d  d  2r0V 1 d
2
  i
 r

2
  dx
r  1  r   dr   dr  
1 d
2 
  i
.............................(5)
 dx 
2Vr0


1 d  d 
2
2
 r
   i 1  r    0......(6)
r  dr   dr  

分離變數求(5)    ci e




  i 2
x 

2Vr0 

the first three eigenfuctions  i are shown by the corresponding eigenvalues.
 0  2.705 , 1  6.667 ,  2  10.67
2

x r
2 
 i  .............(7)
t t   Ci exp    i
2Vr0 r0   r0 
i 0

56
to be determined by B.C at x=0
i
1.0
0
1
2
1.0
 q0 
0
r 
r
r0
2k T0  T    1 d i (1)   i 2 x 

 Ci dr   e
r0

i 0  2
x   x  / Re Pr
x   x / r0
i
i 2
Gi  
0
7.312
0.749
1
44.62
0.544
2
113.8
0.463
3
215.2
0.414
4
348.5
0.382
Ci
 i (1)
2
The mixing-Cup temp
57
tm 

T0  Tm
 1 d i 
  2 x  / i 2 / 2 
 4   Ci
 e i
T0  T
2
dr   r 1
i 0 
 Nu 
G e
 i 2 x 
i
hD
i

..................(8)
2 
2
k
2 Gi e   i x /  i 
i
where x  
x / r0
..........................(9)
Re Pr

Gi  0.5Cii (1)...................(10)
for large x  ; the series becomes increased more convergent until finally for
x   0.1 only the first term is significant
Nu 
G0 e   0
2 
x
2(G0 /  0 )e
02 x

 02
2

7.312
 3.658....................(11)
2
Ref 2-3 ( D) N u  3.658
Thus, the thermal entry length must be approximately
x   0.1
x / r0
 0.1
Re Pr
x
x

 0.05 Re Pr
D 2r0
oil : Pr  100
Re  500
x
   2500
 D  fiD
eq Air Pr  0.7 ; Re  500
T0
T (r1 , x)
58
Tm (x)
x
T
T0
T0
T0
Developing thermal develop
Eq(11)
Nux
Entrance length
Fully develop region
3.658
x
0.1
entry length for developing flow
(B) Solution for uniform heat flux
The uniform – heat – flux, thermal – entry – length problem for laminar flow in a
circular tube is very similar to the constant – surface – temp problem (i.e. Part
(A)).
The same energy Eq with different B.c must be solved by the method of –
separations of variables and Sturm – Liouville theory
We obtain an eigenvalue solution , which is presented here only in terms of the
local Nusselt no

 1
1 e rm x 
Nux  
 
4 
 Nu 2 m Am rm 
2
1
59
where
rm  eigenvalue
at
x
;
Nux  Nu 
48
 4.3 6 4
11
(Thermal fully developed )
B.c
(1) x   0
;
T  T
( x  0)
( 2) r   1
;
q 0  cons tan t
(t   1)
t m
 cons tan t
x
(r  r0 )
T
Fully develop
T0  T ( x)  c o n s
Tm (x)
T
entry region
x
Table
x
Nu x

0
0.002
0.004
0.010
0.020
0.040
12.0
9.93
7.49
6.14
5.19
0.10

4.51
4.36
收斂
60
Extra Notes： Thermally developed
L.F
Laminar Slug Flow in a circular Duct
Pr 

 1

U  c o n tsa nt
For constant – wall temp case

Energy eq： 
Set


1   T 
T
r
 U
r r  r 
x
t
negdecting axial conduction 
Tw  T
Tw  T
1    

r
 U
r r  r 
x
B.c
(1)r  r0
;
(2)r  0
(3) x  0
 0
;  is f i n i t e
;  1
By sep of variables   Rr Z x
U Z 
2
 2x / U




Z

e

 Z

 1 d  r dR   2 R  0  Rr   C1 J 0 r   C2Y0 r   Bessel function

 r dr  dr 
since Y0 0    ;
r  r0    0
C2  0
 J 0 n r0   0
n : e i g e n v as l u e

   C n J 0 n r e 
x / U
2
n
n 1

x  0 ;   1  1   C n J 0  n r 
n 0
r0
 r J  r dr
0
Cn 
0
 rJ  r dr
2
0
0
r0 J 1 n r0 
n
0
r0
n

n
2
r0

J 1 n r0 
2

2
n r0 J 1 n r0 
61
2 n 2x / U J 0 n r 
e
J 1 n r0 
n 0 n r0

 r , x   
T T
2
m  w m  2
Tw  T r0

 4
n 0
e

r0
0
4
rdr 
r0
1

 
n 0

n
2
 r0
e n x / U / J 1 n r0    rJ 0 n r dr
 0
  n x / U
2
n 2

2 2x / U J 1 n r0 
 T 
  
Tw  T 
qw  k 
e
  k Tw  T    k 
J 1 n r0 
 r  r0
 r  r0
n  0  n r0
 k Tw  T 
2
r0

e 

x / U
2
n
n 0

qw
qw
2k  n 2x / U
e  n x / U
1
h

w 
/ 4
e
Tw  Tm Tw  T
r0 n 0
n 2
n 0
2
h2r0   n 2x / U  e  n x / U
 Nu 
 e
/
k
n 2
n 0
n 0
2
n0
N u 
1
 
2 1
 0  2.4048  5.7831
2
2
0

J 0 n   0...................eigenvalue
0  2.4048 ; 5.5201
62
Chapter 3 Natural Convection (Free)
3-1 Boundary Layer Eq for Natural Convection
(A) physical considerations
Natural Convection:
Fluid motion is due to buoyancy forces within the fluid
Buoyancy is due to the combined presence of a fluid
density gradient and a body force that is proportional to
density
(gravitational force usually sometimes a centrifugal force in
a rotating fluid machinery )
heated wall
heating expansion 
compression.
Cold fluid in statics
T0
g
colled floe
cold fluid in statics
  , T
(B) Boundary Layer Eq for Natural Convection
y

velocity B.L
v
H
T0
g
  , T , u  0, v  0
u
(in statics)
x
63
steady – state 2-D flow – with constant properties continuity Eq:
C
p
, k,  
conservation of
m a ss
u v

 0................................(1)
x y
momentum eq :
x  dir
 u
u 
P
 v   
  2 u..............(2)
y 
x
 x
  u
 v
v 
P
y  d i r   u  v   
  2 v  g. . . . . . . . .3.) (
y 
y
 x
energy eq :
u
 T 2 T 2 
T
T
v
   2  2 ........................(4)
x
y
y 
 x
Scale Analysis
 or  T  H  c h a r a c st ot irlcie n g t h
 x ~  or  T

y ~ H

 
 
2  
2  

;

y
x
y 2
x 2
(1)

From(2)

(3)

v  u
P
 0 P  f ( y) o n l y
x
P d P d P


  g
y d y d y
 v
v 
 2v
 v    2      g........................(6)
y 
x
 x
  u
T
T
 2T
v
  2 ............................(7)
x
y
x
Introducing (Define) the coefficient, of thermal expansion
(4)

u
 
1  
 ......(8)
 T  p
64
2

T  T    2 T  T 2 / 2!        .....................(9)
T
T
在 T 展開 Taylor series
   
      T  T ;       T  T ..................(10)

(6)
u
v
v
 2v
 v   2  g T  T ..............(11)
x
y
x
   / 
coupled:
Note: Natura Convection 問題.速度與溫度為聯立方程式求解不像
Forced Convection 問題可以先解出速度再帶入解溫度
3-2 Laminar Natural Convection from a constant Temp vertical Flat plate
(A) Governing Eq’s and B.C ’s
u v

 0. . . . . . . . . .1.). . (
x y
u
v
v
 2v
 v   2  g T  T . . . . . . . . . 2. ). (
x
y
x
u
T
T
v
  2T
x
y
2 
 2T
x 2
B.c’s
(i) Isothermal wall and no – slip condition
u  v  0 ; T  T0 at x  0........................(4)
(ii) v  0 and T  T as x  ....................(5)
(B) Scale Analysis
(3) Expresses a balance between convection and conduction

T0
T
v
65
u
T
; v
T
T
T
~  2 ……………….(11)
H
T
T  T ~ T  T0  T
u
(1) 
T
~
v
Hu
v~
..............................(12)
H
T
  ~ T
H
(iii)  v ~ 2 .......................(13)
T
v
v
v
(2)  u
: v or 2 ~ gT .........................(14)
T
H T
14
H
: 
gT   T
4

H
 Ra H 1 Pr 1 ; 

 T
4

 Ra H 1 ;14

vanishes as Pr  1.......................(15)
where RaH 
gTH 3

......................(16)
(I) Pr  1
(15)

 T ~ HRaH 1 / 4 ............(17)
(13)

v~

H
q0 ~ k
RaH
T
T
1/ 2
...................(18)
 hT
 h~
k
T
Nusselt number
Nu 
hH
1/ 4
~ Ra H .  (Gr Pr)1 / 4 ..........................(19)
k
v
H
T
T


x
66
T  T ~ T  T0  T
(i)
u

~
v
Hu
v~
.......................(12)
H
T
 ~  T
(ii)  v ~
(2)  u
v
T
H
T 2
..........................(13)
, v
v
v
or 2 ~ gT ...............(14)
H T
Next considering effect of inertial friction force
(18)  v
v
v
~  2 ...................(20)
H

 ~ HRa H 1 / 4 Pr 1 / 2 ...............(21)
or

~ Pr 1 / 2  1. . . . . . . . 2. .(2)
T
(II) Pr  1 (inertial effect becomes significant Bejain inertial ~buoyance)
 T ~ H RaH Pr 1 / 4 ................(23)

1/ 2
RaH  Pr  .................(24)
H
1/ 4
N u ~ RaH  Pr  ..................( 25)
v~
The Bousinesque number
Bo H  RaH  Pr 
gTH 3
2
...................(26)
buoyance ~ friction
v
gT ~  2 .........................(27)

1 / 4
 ~ HGrH .........................(28)
where Grashof number
GrH 
gTH 3 Ra H

........(29)
Pr
2
From (22) and (28)

1/ 2
~ Pr  1
T
physical meaning:
67
GrH ~
buoyance
viscousforce
H: wall height
Ra H
1/ 4
Bo H
1/ 4
wall height
if Pr  1
thermal B.L thickness
wall height
~
if Pr  1
thermal B.L thickness
~
(C) Integral Solution
Integrating (2) and (3) from x=0 to x=z  or  T 
z
d z 2
v 
  v dx  v   g  T  T dx........(41)
x  x 0
 dy 0
0
 z
 d vT  T dx   T  ..........................(42)

0
 dy  
x  x 0
 0
(I) Pr  1 C
By x  0 ; T  T0
; x   ; T  T
T
v
T
x
We may assume
T  T  Te  x / T ..........................(43)


v  Ve  x /  1  e  x /  t .....................(44)
sub (43) and (44) into (41) and (42) yields
2

V 2P2
VP2
d 

 gT ...................( 45)


dy  22  P2 1  P2 

P2
 
d 
V 2

  ...................( 46)
dy  22  P2 1  P2  
whereP2 

.....................................( 47)
T
68
A force balance statement n the no – slip layer , 0  x  0  ;
The inertial terms of (2) are identically zero
 2v
 0   2  g T0  T .........................(48)
x
1
P 
5 2 2 2
................................(49)
 Pr  P2
6
P2  2


3


q0 y
P2
3


N u( y) 

1
T0  T k  8


 P2  1 P2  2 P2  2


Specially Pr  1
(49)

Pr 
1/ 4
Ray
1/ 4
.......(50)
5 2 P2
P2
6
P2

6 1/ 2
 P2  Pr ..............(51)
T
5
Similar to (22)
 3
Nu   
8
(50)

(II)
Pr  1
1/ 4
Ray
1/ 4
..................(52)
temp profile (43)
v  V1e  x / T 1  e x /  .....................(53)


(41),(42) and (48)
2
5 P 
 Pr   1  ...........................(54)
3  1  P1 
 P1 
 3
Nu   
8
1/ 4

...................................(55)
T
 P1 


 2 P1  1 
Specially , Pr  1
1/ 2
Ray
1/ 4
.............(56)
2


 Pr  5  P1   P1  1


3 1 


69
 3 
  Pr  ......................(57)
T  5 
1/ 2
(54)

P1 
N u  0.689
P R 
1/ 4
r
ay
...................(58)
3-3 Natural Convection along a vertical plate with Uniform wall Heat flux
(A) similarity solution
Governing Eq:
u v
Continuity:

 0......................(1)
x y
Momentum: u
Energy: u
v
v
 2v
 v  g T  T    2 .............(2)
x
y
x
T
T
 2T
v
  2 ..................(3)
x
y
x
B.c’s
x  0 ; u  0 ; v  0. . . . . . . . . . . . . . . .4.a.). (
T
 q 0 / k .....................(4b)
x
x 
im v  0.....................(5a)
x 
im T  T ...............(5b)
x 
v  0..........................(5c)
y  0
T  T .......................(5d )
70
 3-3 Natural Convection Along A Vertical Plate With Uniform Heat
Flux
Y,v
G
T  T ; u  0
X,u

(A) Similarity solution
Governing EQ:
u v
Continuity:

 0 ……………………………………………….. (1)
x y
Momentum: u
Energy: u
u
v
 2v
 v  gB(T  T )   2 ………………………... (2)
x
y
x
T
T
 2T
v
  2 ………………………………………... (3)
x
y
x
BC’s
u  0 : v  0

x  0;  T
………………………………………………….....(4a,b)
 x  q 0 / k
lim v  0
x 
x  ; 
……………………………………………………...(5a,b)
T  T
lim
x 
v  0
…………………………………………………………..(5c,d)
y  0; 
T  T
Introducing a similarity variable “  ”
u

y
;
v

…………………………………………..(6,7)
x
 ( x, y )   ( y ) f ( ) …………………………………………………….(8)
T  T  Y ( y) ( ) ……………………………………………………...(9)
  xG( y ) ……………………………………………………………...(10)
71
Form (4b) Y ( y)G( y) (0)  q0 / k
 Y ( y) 
 q0
 q0
[ (0)] 1 
kG( y )
kG( y )
From (9) T  T 
 q0
 q 0  ( )
 ( ) 
………………………..(9’)
kG( y )
k G( y)
G
f  ………………………………………….(11)
G
From (7) v  Gf  ……………………………………………………...(12)
From (6)  u   f  
Where:   
d ( y )
dG
df
; G 
;f
dy
dy
d
Hence,
From (2) f  
From (3)   
q Bg 

G  
 
2
ff   
 2  f   0
 0 ………….(13)
G
k G 4
 G G 

G 
Pr f  
Pr f   0 ……………………………..(14)
G
G 2
Where: Pr   / 
In order to make (13)(14) became O.D.E we need to assume (for similarity)

(a)
q0 Bg 1
 a1  1
k G 4
(assume=1)
q0 Bg 1
…………………………………………………(15)
k G 4
G 
(b) 2  a 2
(assume=constant)
G
 

Sub. (15a) into (15b)
 G 6
G
k 2
 a2
y
q 0 g
 G 5  5a 2

(c)
1
k 2
  G 5  a 2
y  c1
5
q0 g
k 2
y ……………………..general solution
q0 g

 a3
G
y  0; G  , T  T
a2  1
72
 q g 
 G ( y )   0 2 
 5k y 
1/ 5
………………………………………………..(
16)
 q g 
   x 0 2 
 5k y 
1/ 5
…………………………………………………..(
17)
Sub (16) into (15) yield:
 q g 
  ( y )  5  0 2 
 5k y 
1/ 5
y 4 / 5 ………………………………………….
(18)
Condition (c)

 a3  4
G
From
(13) f   4 ff   3( f ) 2    0 ………………………………(19)
From
  0 …………………………………(20)
(14)    4 Pr  f  Pr f 
V  Gf   5
uv
3/ 5
3/ 5
v
 q0 g 


 k 
1/ 5
1/ 5
 q0 g 


 k 
2/5
y3/ 5 f 
 1 
(f   4 f )

1/ 5 
 (5 y ) 
1
q
T  T  0
k
 5kv2  5

 
 q0 Bg 
NOTE: Tw  T   e y
or
q
0

 e y  still exact similarity
solution

Refer to: Kays
(B)Scale Analysis
q0  k
T
………………………………………………………………….(31)
T
For high Pr (Pr>>1)
73
From (17) in
 gBTH 3 

chapter”2”  T  H 
 v

1 / 4
………………………………(32)
From (31)
(32)  T  H ( Rah ) 1 / 5 ……………………………………………...(33)
gBH 4 q0
Where:
 Ra H ………………………….(34)
kv
T 
q 0 T q 0

H ( Ra H ) 1 / 5 …………………………………………………....(
k
k
35)
Nu 
q0 y 0
………………………………………………………………..(3
T0  T k
6)
 Nu 
H
T
 Ra H
1/ 5
………………………………………………………(37)
(I)  T  y 1 / 4 , T  const. q0  y
1
4
as
isothermal wall
Y,v
(II)  T  y 1 / 5
1
q0  c o n .s t T  y 5 as
isothermal wall
X,u
For low Pr (Pr<<1)
From (23) in
chapter”2”  T  H Ra H Pr 
1 / 5
 T 
……………………………….…(38)
q0
H ( Rah Pr) 1 / 5 …………………………………………………….(39)
k
74
 Nu  ( Ra h Pr)1 / 5 …………………………………………………………(40)

(C)Integral Analysis (Ref. SPARROW)
2
Nux 
3601 / 5


 Pr 

 Ra xy 1 / 5
4


  Pr 
5


 Extra Notes
Similarity Solution:
Natural Convection on an isothermal vertical flat plate
X,u
U  0
T=constant
u

 

u  y ; v  y


  T  T

T0  T
Y,v
MOMENTUM:
ENERGY:
&
  2   2
 3


gB
(
T

T
)


v
0

y xy x y 2
y 3
   
 2

 2  0
y x
x y
y
  yH (x)
Assume  ( x, y )  vf ( )G ( x)
Grx 1 / 4

G ( x)  4( 4 )
gBx 3 (T0  T )
when 
; Grx 
v2
 H ( x)  1 ( Grx )1 / 4

x 4

 f     3 ff   2 f  2  0........................................................................(i)

   3 Pr f   0.....................................................................................(ii )
BC: y = 0; u = 0; f’ (0) =0; f (0) =0
75
 y  ; u  0; f ' ()  0

 y  ; T  T ; ()  0

 y  0; T  T0 ; (0)  1
Nux 
xhx
 (0) 1 / 4
  (0) xH ( x)  
Grx
k
2
Numerical solution
1/ 4

3
2 Pr
 Nux  
1/ 2
4  5(1  2 Pr  2 Pr) 
Grx Pr 
1/ 4
1/ 4

3
2 Pr
 
1/ 2
4  5(1  2 Pr  2 Pr) 
T

0   k (T0  T )
x
y
Where:
q 0  k (T0  T ) (0) H ( x)
q0  k
0
 k (T0  T )
 
 y
Ra x 1 / 4
0
Similarity Solution:
Natural Convection with variable surface temperature
Consider the power-law variation
X,u
U  0
T=variable
u
Y,v

 

u  y ; v  y

T  T  Ax n
n0

 0
T0  T  A
Assume  ( x, y )  vF ( )G ( x) ;   yH (x)

 f     (n  3) ff   (2n  2) f  2  0............................................................(i)

 ]  0......................................................................(ii )
   Pr[( n  3) f   4nf 
BC:
F (0)  F ' (0)  0; (0)  0
F ' ()  0; ()  0
76

q 0


q 0


q 0


q 0
 
0
 y
1
 k (T0  T ) (0) H ( x)  k (T0  T ) (0)
Grx1 / 4
2x
 k
T
x
0
 k (T0  T )

y
0
 k (T0  T )
x3/ 4
 g    (0)
 k 2 
(T0  T ) 5 / 4
x
2
v 
 C (T0  T ) 5 / 4 x 1 / 4
1/ 4
1/ 5
Pr


 Nux  

1/ 2
 4  9 Pr  10 Pr 
Gr Pr 
*
x
1/ 5
;
Grx  Grx Nux 
*
g
q x4
2 0
kv
77
Chap4 Convection with Change of Phase
 4-1 Condensation
Physical Mechanisms
Modes of condensation
(a) Film condensation
(b) Drop wise condensationExcellent effect
(c) Homogeneous condensation
(d) Direct contact condensation
78
 4-2 Film Condensation
Flow regions of film condensation on a cooled vertical surface
79
Ref:
Dhir V.K. and J.H.Lienhard [Laminar Film Condensation on Plane
and Axi-sysmmetric Bodies in Non-uniform Gravity]
80
 (A) Film condensation for isothermal vertical plate
-In the condensation boundary layer for steady state flow
Governing EQ:
u v
Continuity:

0
x y
 u
v 
P
 2u
Momentum:  L u
v   
  2  l g
y 
x
y
 x
2
T
T
 T
Energy: u
v
 2
x
y
y
Outside boundary layer (vapor zone)
P
 v g
x
The vertical pressure gradient in the liquid is the same as the hydrostatic pressure
gradient in the outside vapor.
 u
v 
 2u
 l u  v     2  (  l   v ) g ……………………………..(2)
y 
y
 x
Assumption by Nussle [1916]
(1) Laminar flow and constant properties are assumed for the liquid
film.
(2) The gas is assumed to be a pure saturated vapor at Tsat.
(3) The shear stress at the liquid –vapor interface is assumed to be
u
neglible in which case
y   0
y
81
(4) Momentum and energy transfer by advection in the condensation
film are assumed to be neglible (because the velocity of film is so
small ,we can neglect inertial term compared with frictional sinking
forces)
 2u  g

(  l   v ) …………………………………………….....(3)
y 2  L
(1) y  0; u  0; T  TW
B.C
(2) y   ;
u
y

 0, T  Tsat
Integrating w.r.t y yield
 y 1  y 2 
u ( y )  (  l   v )      ………………………………(4)

  2    
g
2
In which the film thickness is an unknown function of longitudinal position
(  ( x)  ? )
The local mass flow-rate through a cross-section of the film
*
 ( x)
m( x)
  u ( y)dy  ( x) ……………………………………….(5)
0
b
Lengths is the direction normal the plate
(dA=bdy)
82
Considering the control Volume  ( x)dx shown in fig .the steady –state from of the
energy eq.(1st.law) can be written as follows
g
( x)  l (  l   v ) 3 ……………………………(6)
3
Enthalpy inflow
associated with the
mass flow-rate (m)
-
Enthalpy outflow
associated with the mass
flow-rate (m+dm)
Enthalpy inflow through
interface associated with
+
saturated vapor (dm)
The heat conducted through
-
the wall on heat flux
absorbed by the wall
 ( x)
H 
0
=0
H+(H+dH)-q(bdx)=0
u(h f  C Pl (Tsat  T ))dy ………………………………(7)
Since the liquid is slightly sub cooled T<Tsat (so,h<hf)
H = hf – CPl (Tsat - T)
Nusselt assumed the local temperature T is distributed linearly approximately
across the film.
 y  0; T  Tw
T T
 2T
y
Because
 sat

0
;
 1  ……………..(8)

2
Tsat  Tw

y
 y   ; T  Tsat
After using (4) and (8) in the integral (7) .we obtain
3


H  h f  C Pl (Tsat  Tw ) ( x) ………………………………………(9)
8


Further, by the linear temperature profile across the film
T  Tw
T
q w  k
 K sat
……………………………………………..(10)
y
 (x)
Energy balance
H  ( H  dH )  hg d  q w dx  0 ………………………………………..(11)
After using (9) and (10), we have
k
3


(Tsat  Tw )dx  hg  h f  C Pl (Tsat  Tw ) d  hfg d ………….(12)
 ( x)
8


NOTE: Rohesnow recommend using nonlinear temperature profile across
83
the film
hfg  h fg  0.68C Pl (Tsat  Tw )  h fg (1  0.68 Ja)
； Ja 
C Pl (Tsat  Tw )
h fg
h fg is the augmented latent heat of condensation which includes the proper
latent heat (hfg) and a contribution accounting for the cooling of the fresh
condensate to temperature below Tsat.
Differentiating eq. (6) yields
 g (    v ) 2  ( x)
…………………………………………………..(13)

x

x
Combining eq. (12) and eq. (13) .It follows that
k (Tsat  Tw )
 3 d 
dx …………………………………………………….(14)
g (    v )hfg
Integrating from x = 0 (  = 0) to any x location of interest on the surface
 4 k (Tsat  Tw ) x 
 ( x)  

 g (    v )hfg 
1/ 4
…………………………………………………...(15
)
The surface heat flux may be expressed as
k

qW  h(Tsat  Tw ) 
(Tsat  Tw ) ………………………………………….(16)
 ( x)
Convection heat transfer coefficient ：
h
k
…………………………………(17)
 (x)
From (15)
 g (    v )hfg k 3 
h

 4 (Tsat  Tw ) x 
1/ 4
………………………………………………………(1
8)
hx x
k
The average h
Note: Nu x 
 g (    v )k 3 hfg 
1 L
h   hx dx  0.943

L 0
  (Tsat  Tw ) L 
1/ 4
………………………..………….(1
9)
The average Nusselt number then has the form.
84
Nu 
hL
 Ra 
 0.943 
k
 Ja 
1/ 4
3
Ra Gr Pr  (    v ) g (hfg ) L


Ja
Ja
k (Tsat  Tw )
; Gr 
Buoyancy  Force
;
Viscous  Force
enthalpy  Flows
Conduction
C (T  Tw )
Ja  P sat
h fg
Pr 
 (B) Film Condensation on Radial system(axi-symmetric bodies
surface)
R(x)
Tsat  T
Vapor
film
y
x
R = radius of curvature
As R 
= vertical plate
R= constant =sphere
For isothermal temperature (Tw=constant) surface assuming that the condensate
film is inertia less (thin film), & non-vapor drag & non-condensable (inert gas)
gas.
Momentum:
 2u  g

(  l   v ) ………………………………………….(1)
l
y 2
(1) y  0; u  0; T  Tw
B.C
(2) y   ;
u
y

 0, T  Ts a t

y2 
u 
(    v )  ( x) y   ……………………………………………(2)

2

g ( x)
85

 2T
 0 ……………………………………………………………………..(3)
y 2
y
 Tsat  T  (Tsat  Tw )(1  ) ……………………………………………….(4)

Mass flow rate of condensate

    udy 
0
g ( x)(    v ) 3
……………………………………………….(5)
3
(0 ~  )

CV 

 x  x  dx 
Energy balance

qW (2Rdx )  hfg d (  2R)
k
T
y
y 0
   v  3 
 
g ( x)
 R( x) ………………………………(6)

x 
3

R   hfg
Where：
hfg  h fg  0.68C P (Tsat  Tw ) …………………………………………(7)
Sub.(4) into (6)
k

    v   3
 R( x) g ( x)………………………………..(8
3R( x) x
(Tsat  Tw )dx  hfg
)
 *  g 1 3 ( x) R1 3 ( x) ……………………………………………………………….
(9)
 * hfg  (    v )  *3
kT 
( ) …………………………………………………..
3R 4 / 3 g 1 / 3 x
(10)
by separation of variables：
Eq. (10) can be integrated
x
4kT
( * ) 4 
R 4 / 3 ( x) g 1 / 3 ( x)dx

hfg  (   V ) 0

4kT
 
 hfg  (    v )

geff 

x
0
R 4 / 3 ( x) g 1 / 3 ( x)dx 

R 4 / 3 ( x) g 4 / 3 ( x) 

1/ 4
R 4 / 3 ( x) g 4 / 3 ( x)
1 x 4/3
R ( x) g 1 / 3 ( x)dx

0
x
86
 g eff ( x)  (    v )hfg k 3 
h 

 
4 (Tsat  Tw ) x

1/ 4
k
Note: Nu x 
hx x
k
3
h x x  g eff ( x)  (    v )hfg k 3 
Nu x  x   
x 
k
 
4 (Tsat  Tw )

1/ 4
The average h
h
1 L
hx dx
L 0
The average Nusselt number then has the form.
1/ 4

hL
 Ra  

Nu 
 function  
k

 Ja  

3
Ra Gr Pr  (    v ) g (hfg ) L


Ja
Ja
k (Tsat  Tw )
Pr 
Ja 
; Gr 
Buoyancy  Force
;
Viscous  Force
enthalpy  Flows
Conduction
C p (Tsat  Tw )
h fg
 (C) Laminar Film Condensation with Heat Generation on
Vertical Plate
For steady-state film condensation of a heat-generation fluid on a vertical
surface neglecting inertia terms, the momentum (or velocity) boundary layer
is
87
Q
Momentum:
 2u  g

(  l   v ) ………………………………………….(1)
y 2  (l )
(1) y  0; u  0; T  Tw
B.C
(2) y   ;
u
y

 0, T  Ts a t
(zero shear stress or no vapor drag)
Integrating directly (1) yields.
2
g(  v ) 2  y 1  y  
u
      ……………………………………………..(2)
v
  2    
The energy eq. for the heat –generating film, neglecting convection is given by
the conduction eqn.
k
 2T
 Q …………………………………………………………………...(3)
y 2
Where: Q is the volumetric heat generation rate.
Integrating (3) yields
Q 2
T  Tw 
2k
 y  2 y 
y
     T ……………………………………………(4)

    
equating the heat conduction into the film with the heat liberation by
condensation at interface
T

k
( u ) ……………………………………………………..(5)
  h fg
y
x
88
1

Where u 

0
udy ………………………………………………………..(6)
Ref: 1975 J.Heat Transfer p.141~142
Sub u into (6) leads to
1 g 2 (    v )
…………………………………………………………(7)
3
v
Combining eq(4)(5) and (7) yields the O.D.E
u
 2 1   2
…………………………………………………………………(8)


2
1/ 2

 Q 

  
 2kT 

…………………………………………(9)(10)
where
vQ 2 x
 

2 g (h fg )kT (    v )

Integrating of eq. (8) with the B.C
 (0)   (0)  0
Gives


   ln 1   2   2 ………………………………………………………….(11)
the local heat transfer of the vertical wall q w follows from differentiating eq.(4)
and is
(i.e.： q w  k
 q w  

qW
kT


2QkT 
1/ 2
T
y
(
0

kT


Q
)
2
Q 1/ 2 Q  Q 
)  
2kT
2  2kT 
1 / 2
1  QkT 
 

 2 
1/ 2
 kTQ 


 2 
1/ 2
)

11
     
2 

(D) Forced convection film condensation on a cylinder
T
u
89
Fig1.
Fig2.
For a laminar , steady-state condensate film with constant fluid properties and
usual Nusselt’s (1916) assumption.
The governing equations are：
Mass equation :
m  

d
u  dy ……………………………….….(1)
rd 0
Momentum:

 2u
dp
    v g sin  
 0 …………………...(2)
2
rd
y
With Boundary conduction：
[1] y=0； u  0 …………………………….(3)
90
[2] y=  ； 
u
   …………………………(4)
y
Energy equation :
h fg m   k
T
y
 kT
 =
………………………………(5)

0
By shekriladge and Gomelauri (1966) model for simplicity to the interfacial vapor
drag:
   m  ue ………………………………………...…(6)
Where : u e = the vapor velocity at the edge of boundary layer

u   mu e


………………………………..(7)
y 

Bernoulli equation for vapor :
du
dp

  v u e e ………………………………..…..(8)
rd
rd
The resultant momentum equation by sub. (8) into (2) and applying boundary
conduction (3).(7) can be integrated as
mu e y 
du 
1
u
 (    v ) g  sin    v u e e  ( y  y 2 ) /  …………..…..(9)

rd 
2

According to potential flow theory
u e  2u  sin  ……………………………………….(10)
B.L.
ue
u
Assuming  << r, the pressure grad. Can be
du
1 dp

 u e e  4  v u 2  sin 2 / D ………………(11)
r d
rd
Sub. (11) into (9) , and sub the resultant velocity u into (1) , and sub the updata m
into (5) , we finally obtain the IO.D.E ：
2 *

 v Pr * 4
d  *
1
2
*3
 sin    1 …….(13)
 sin   ( Re  Ra / J a ) sin  
dp 
3
 Ja
3

91
 
 * =   Re
D
Where
   ;
c
*


Where
Re  Du  / 
Ra   (    v ) g Pr D 3 /  2
Ja 
Nu 
D

Nu  Ra

1 2
Re
*

c
h fg
……………………………………..(14)
1

C p T
*
d /  ……………………………..(15)
0
(E) Turbulent film condensation
fig4.(flow region)
92
(x) : condensate rate for plate of width b .
The transition criterion is Reynold number (in term of  ):
4
Re  
 (l )
recall:
Re d 
vD

 m  VA  V
 Re d 
4


D 2
4
VD

The condensate mass flow rate :
m  u b  b  
   u 
u
1

Re  

0
u  dy
4 u 

At , Re  =1800 , the transition from laminar to turbulent flow is complete.
For the wave free laminar region Re   30 , and it’s  (x ) can be obtained.

   udy 
0
1     v g 3
3

93
 Re  
4 g    v  3
3 l
   v
Assuming :
Re  
4 g 2  3
3 l
………………………………..(1)
2
2
=
4 g 3
3 l
由壁所帶走之熱值須等於質量流乘以蒸汽凝結潛熱：
T  Tw 
    v g 2
d  hfg  k  dx  sat
l

   v
 4k l  l T  x 
 ( x)  

 g    v h fg 
h( x ) 
1/ 4
 4k  T  x 
  l l2

 g hfg 
1/ 4
k
 ( x)
平均值：
l
 g 2 k l 3 hfg 
1
h   h( x)dx =0.943 

l0
  l T  l 
From (1).(2).(3) and can cleaning :



1/ 4
T
h fg
h (u l /  ) 2 / g
1 / 3
…………………………… Re   30
 1.47 Re 
kl
In the laminar wavy region , Kutaledgz[1] recommends a correlation of the form :

h l / g
kl
2

1/ 3

Re 
1.08 Re 
1.22
 5 .2
;30  Re   1800
For the turbulent region , Labuntson [2] recommends
h ( l / g )1 / 3
Re 
; Re   1800

0.75
0.5
kl
8750  58 Pr (Re   253)
2
Ref[1]:
Kutaladge S.S.”Fundamentals of heat transfer “ Academic press , New York ,
1963
Ref[2]:
Labuntsou,D.A. “heat transfer in film condensation of pure steam on vertical
94
surface and horizontal tubes “ teploenergetika , Vol4, pp. 72, 1957
 4.2 Boiling
Boiling Modes：
(A)
Definition: Boiling
When evaporation occurs at solid-liquid interface, it is termed Boiling
The boiling process occurs when Tw  Tsat
(P)
By Newton’s Law of cooling
q w  h(Tw  Tsat )  hTe
Two types of modes
(a) Pool boiling：
The liquid is quiescent and it’s motion near the surface is due to free convection
and to mixing induced by turbulent growth and detachment.
(b) Force convection boiling：
Fluid motion is induced by external means as well as by free
convection and bubble-induced mixing.
(B)
The boiling curve (in pool boiling)
95
C
B
D
A
Fig1. the four regimes of pool boiling in water at atmospheric pressure.
A: Onset of nucleate boiling
A B: Isolated bubbles
B C: vapor escapes are jets or columns
C : burnout point (Critical heat flux point)
fig2. the pool boiling curve in a temperature-controlled experiment (left side ) , and in
a power-controlled experiment (right side).
Parameters could shift the boiling curve：
(1) Surface characteristics (material property of heater )
a. roughness
b. surface chemical effect
oxidation, cleaness , eta.
(2) Thermal physical properties of the liquid.
(3) Heater geometry
(4) magnitude of acceleration (or gravity )
(5) system pressure (P)
96
(6) liquid subCooling (state of the liquid :Saturated , Subcooling)
(7) Flow velocity (pool or forced flow condition )
Two-phase flow
(C)
Nucleate Boiling
The most important regime of the entire boiling curve is the regime of nucleate
boiling , because here the boiling heat –transfer coefficient
qw
h
Tw  Tsat
Reaches characteristically large values
10 3  h  10 5 ( w / m 2  k )
Rohsenow [32] proposed by his experiment and correlations
Tw  Tsat 
h fg
C p ,l
 q 


Pr l C sf  w (
)1 / 2 
  l h fg g (    v )

1/ 3
s
For water： Pr s  s  1
實驗數據:
water - copper polished C sf =0.013
water – copper scored C sf =0.068
water – platinum
C sf = 0.013
12
 g (    v )   C Pl (Tw  Tsat ) 
q w   l h fg 
  Pr s l  C  h 


 

sf
fg 
(D)
3
Film Boiling and Minimum Heat Flux
97
98

Definition： qmin
 ）is registered at the lowest heater temperature
The minimum heat flux （ qmin
where the film is still continuous and stable.
1.
For a sufficiently large horizontal plane surface
g  l   v 
  0.09h fg   v 
qmin
2 
  l   v  
2.
1
4
For the film boiling on a horizontal cylinder
 D 3 h fg g  l   v  
hd  d
Nu 
 0.62 

kv
 k v v Tw  Tsat  
1
4
hfg  h fg  0.4C Pv Tw  Tsat 
3.
For the film boiling on a sphere
 D 3 hfg g  l   v 
hd  d
 0.67 

kv
 k v v Tw  Tsat  
1
4
Note that, Bromley showed that the thermal radiation effect can be incorporated into
an effective heat-transfer coefficient h
3
h  hD  hrad
4
where hrad 

 w Tw 4  Tsat 4
T
 Tsat
w

（when hD > hrad ）

when hD  hrad
h
h  hD  D
 h



1
3
 hrad
99
Dimensional Analysis
Basic Dimension
(一) [M] [L] [t] [T]
(二)
[F] [L] [t] [T]
where:
[M] :mass
[L]:length [t]:time [T]:temperature [F]:force
Derived Dimensions:
Q : [F] [L] / [t]
(A)
A rational way of investigating a problem is to describe the problem first
in terms of approximate dimensionless number. The development
loading to a dimensionless dissipation of a problem is called
dimensional analysis.
(i) physical similarity:
Ration established from individual terms of approximate general
principles gives the dimensionless grops.
(ii) Term –by –term non-dimensionlization of the governing equation.
(iii) Buckingham’s  theorem:
If a physical quantity Q1 , can be expressed as a function of (n-1)
other independent physical quantities or variables Q2 , Q3 , Q4 ,….
Qn
If we have
Q  f (Q2 , Q3 ,.......Qn ) ……………………………(1)
100
we can reduce the total number , of mathematical variables form
n to n-p , where P is the number of fundamental dimensions required to
describe the physical quantities involved.
Physical quantities dimension
Length [L]
Mass [m]
Time [t]
Temperature [T]
*amount of substance [n] (mole number )
heat of substance [Q] 形成 5 個，Fundamental dimensions.
It is true that fluid friction will always produce some thermal effects ,
but in many cases this is insignificant compared will the heat that is
transferred by conduction and convection.
In problem of this kind, it will be appropriate to introduce the fifth
dimension for heat [Q].
Physical quantities Dimensions
1
2
3
4
5
Heat quantity
Heat transfer rate
Heat flux (q) 單位時間傳的熱量/面積
Heat transfer coefficient (h)
Specific heat ( C p )單位時間所需要升高的密度
6 Thermal diffusity (  )
熱擴散係數
ML2 t 2
Q
ML2 t 3
Q/t
Mt 3
Qt 1 L2
1 2 1
Mt 3T 1 Qt L T
1 1
L2 t 1T 1 QM T
l 2 t 1
l 2 t 1
101
(B) Forced convection heat transfer
N u  f (Re, Pr)
when , considering heat transfer to a find flowing in a circular –pipe
q  f (T , k , C p ,  , D, u m , Tm )
 T
Cp
D
k
Tw
q :heat flux , 熱通量(單位面積熱傳率)
(1)
有 n=9 個變數 quantities
P=5,fundamental dimensions
n-p=4
故有 4 組，  1 ,  2 ,  3 ,  4 無因此參數。
(2) 選取 k ,  , D,  , T 為重覆變數
1 
q
k  Dc  dT e
a
b
a
b
 Q   Q  M 
c
 2 
  3   L 

 L t   LTt   L 
d
M 
e
    T 
 Lt 
比較系數:
Q: a=1
T: 0=-a+e  e=1
t: -1=-a-d  d=0
M: 0=b+d  b=0
L: -2=-a-3b+c-d  c=-1
102
1 
where
q
qD
hD
=
= Nu =
c d e
kT
k
k  D  T
a
b
q
h
T
Similarity :
u  D
 Re

um   2 
C p
Cp  3 
Tm   4 
k
 Pr
Tm
(Nondimensional Temp prof.)
T
N u  f (Re, Pr,
Tm
) ……………………..(3)
T
(3) reduced to
N u  f (Re, Pr) …………………………………(4)
other choose :
(i)
 , C
p
, u m , Tm 
T  1
Nu  f (Pe, Re)
where
Pe  Re Pr
working with [ M ][ L][t ][T ] system instead , we require five primary
quantities N-P=9-4=5 ( Re; Pr; Ee;
Tm
)…………………..(7)
T
at low speed with small temp. difference
Nu  f (Re, Pr) ……………………………………(8)
103
Free Convection
h  f ( L, g ,  ,  , C p , k )
q  hT
故不出現 q, T
n=7,quautities
p=5, primary dimensions [ M ][ L][T ][t ][Q]
n-p  2,獨立的無因次群(independent dimensionless group)
改用 [M ] [L ] [t ] [Q] /[T ] …………………………….4 primary dimensions
理由
h  [ L2 t 1 (Q / T )]
C p  [M 1 (Q / T )]
k  [ L1t 1 (Q / T )]
出 現 Q, T 均 以 Q / T 出 現 ， 故 可 用 7-4=3 ， 無 因 次 群 (dimensionless
group ) h, g , C p 
using Buckingham’s  theorem to derive
Nu  f (Pr, Gr ) ……………………..(10)
where
Gr 
gL3
………………………..(11)
2
1   
     ………………………..(12)
  T  p
for small temperature change    T ……………(13)
from equation (11):
104
Gr  g 2 TL3 /  2 …………………………………………(14)
Buckingham’s  term-
Free Convection
Quantities :
dimension
L
2 1
h
t Q /T
L
g
[L ]


[ ML3 ]

[ML2 t 2 ]
[ ML1t 1 ]
[M 1 (Q / T )]
Cp
[ L1t 1 (Q / T )]
k
無因次群(dimensionless groups)
7-4=3
1 :
h
k    Lc1   d1
a1
b1

 
 L2 t 1 (Q / T )  L1t 1 (Q / T )
 ML  L ML t 
a1
3 b11
c1
1 1 d1
Q
( ) : 1  a1
T
(t ) : 1  a1  d1  d1  0
M  : 0  b1  d1  b1  0
L : 2  a1  3b1  c1  d1  c1  1
1 
2 

hL
 Nu ( L )
k
Cp
k  Lc2  d 2
a2
b2
 
 ML  L ML t 
 M 1 Q / T   L1t 1 Q / T 
a2
3 b2
c2
1 1 d 2
Q
  : 1  a2
T 
t  : 0  a2  d 2  d 2  1
105
M  : 1  b2  d 2  b2  0
L : 0  a2  3b2  c2  d 2  c2  0
2 
Cp
 Pr
k
 C p


k
Pr 
3 
g
k  b3 Lc3  d3
a3

 
 ML  L ML t 
 ML2 t 2  L1t 1 Q / T 
a3
3 b3
1 1 d 3
c3
Q
  : 0  a3
T 
t  : 2  a3  d 3  d 3  2
M  : 1  b3  d 3  b3  1
L : 2  a3  3b3  c3  d 3  c3  3
3 
g    L3

g ( T ) L3

2
 2 gTL3

 Gr
2
 1  f ( 2 ,  3 )
Nu  f (Pr, Gr )
Bucking hum  -Them---forced convection
q  f (T , k , C p ,  ,  , D, u m , Tm )
1  q / k a  b Dc  d T e   1 
1
2 
1
1
1
1
qD
hD

 Nu
kT
k
um
k  D c2  d 2 T e2
a2
 
b2
a
2
 Q  M 
 Lt 1  
  3
 L  t T   L 
b2
Lc
2
M 
 Lt 
 
d2
T e
2
106
Q : 0  a2
T  : 0  a2  e2  e2  0
t  : 1  a2  d 2  d 2  1
M  : 0  b2  d 2  b2  1
L : 1  a2  3b2  c2  d 2  c2  1
2 
3 
u m D

 Re
Cp
k a3  b3 D c3  d3 T e3


a
b
d
 Q  M 
c M 


T e
 QM 1T 1  
L
3





 Lt  T   L 
 Lt 
Q  : 1  a3
(t ) : 0  a3  d 3  d 3  1
(T ) : 1  a3  e3  e3  0
(M ) : 1  b3  d 3  b3  0
( L) : 0  a3  3b3  c3  c3  0
3 
4 
Cp
k
Tm
k  D c4  d 4 T e4
a4
b4
a
4
Tm    Q   M3 
 LTt   L 
b4
L
c4
M 
 Lt 
 
d4
T e
4
Q : 0  a4
[t ] : 0  a4  d 4  d 4  0
T  : 1  a4  e4  e4  1
Tm
()T
 1  f  2 ,  3 ,  4 
4 
T 

Nu  f  Re, Pr, m 
T 

107