Chapter 6
Fundamental Concepts of
Convection
6.1 The Convection Boundary Layers
Velocity boundary layer:
τ
= μ ∂u
s
surface shear stress:
∂y y=0
τs
local friction coeff.: C f ≡ ρ u2 / 2
∞
Thermal boundary layer:
local heat flux: qs′′ = −k f ∂∂Ty
local heat transfer coeff.:
(6.2)
(6.1)
(6.3)
− k f ∂T / ∂y y =0
qs"
=
h=
Ts − T∞
Ts − T∞
y=0
(6.5)
Concentration boundary layer:
local species flux: N A" = − DAB ∂C A
∂y
or (mass basis):
n"A = − DAB
∂ρ A
∂y
local mass transfer coeff.: hm =
or (mass basis):
[kmol/s ⋅ m 2 ]
(6.6)
y =0
[kg/s ⋅ m 2 ]
y =0
N A" , s
C A, s − C A,∞
−DAB ∂ρA / ∂ y y=0
hm =
ρ A,s − ρA,∞
=
− DAB ∂C A / ∂y y =0
C A, s − C A,∞
(6.9)
6.1.4 Significance of the Boundary Layers
„
„
„
Velocity boundary layer: always exists for flow over any
surface
Thermal boundary layer: exists if the surface and free
stream temperature differ
Concentration boundary layer: exists if the surface
concentration of a species differs from the free stream value
The principal manifestations and boundary layer
parameters are
„
„
„
Velocity boundary layer: surface friction and friction
coefficient Cf
Thermal boundary layer: convection heat transfer and heat
transfer convection coefficient h
Concentration boundary layer: convection mass transfer
and mass transfer convection coefficient hm
6.2 Local and Average Convection Coefficients
6.2.1 Heat Transfer
The local heat flux q” may be expressed as (Newton's law of cooling)
q”= h(Ts-T∞), h = heat transfer convection coeff.
= f (fluid properties, surface geometry, flow conditions)
The total heat transfer rate q may be obtained by integration
q = ∫ q′′dAs = (Ts −T∞ )∫ hdAs , if Ts is uniform
As
As
(6.10-12)
= hAs (Ts − T∞ ),
1
where h =
As
∫
As
hd As .
(6.13)
6.2.2 Mass Transfer
Similar for mass transfer of species A, we have
N A" = hm (CA, s − CA,∞ ) [kmol/s ⋅ m 2 ] --local --Molar transfer rate
(6.15)
N A = hm As (CA, s − CA,∞ ) [kmol/s] --average
(6.16)
where hm = ∫ hm dAs
As
or n" = h ( ρ − ρ ) [kg/s ⋅ m 2 ]
A
A, s
A,∞
m
nA = hm As ( ρ A, s − ρ A,∞ ) [kg/s]
--Mass transfer rate
(6.18)
(6.19)
We can also write Fick’s law on a mass basis by
multiplying Eq. 6.7 by MA to yield
∂ρ A
n = − DAB
∂y
= hm ( ρ A, s − ρ A,∞ )
"
A
hm =
(6.20)
y =0
− DAB ∂ρ A / ∂y y =0
ρ A,s − ρ A,∞
(6.21)
The value of CA,s or ρA,s can be determined by assuming
thermodynamic equilibrium at the interface between the
gas and the liquid or solid surface. Thus,
CA, s
psat (Ts )
=
RT
(6.22)
6.2.3 The Problem of Convection
The local flux and/or the total transfer rate are of
paramount importance in any convection problem. So,
determination of these coefficients (local h or hm and
average h or hm ) is viewed as the problem of convection.
However, the problem is not a simple one, as
h or hm = f (fluid properties, surface geometry, flow conditions)
↑ i.e., ρ , μ , k f , c p , f
EXs 6.1-6.3
6.3 Laminar and Turbulent Flow
Critical Reynolds no. where transition from laminar boundary layer
to turbulent occurs: Re = ρ u∞ xc = 5 ×105
x,c
EX 6.4
μ
6.4 The Boundary Layer Equations (2-D, Steady)
6.4.1 Boundary Layer Equations for Laminar Flow
For the steady, two-dimensional flow of an incompressible
fluid with constant properties, the following equations
are involved. (Read 6S.1 for detailed derivation.)
∂u ∂v
„ continuity eq.:
+
=0
∂x
∂y
(D.1)
„
momentum eq. (incompressible):
x-dir.
y-dir.
„
⎛ ∂ 2u ∂ 2 u ⎞
⎛ ∂u
∂p
∂u ⎞
ρ ⎜⎜ u + v ⎟⎟ = − + μ ⎜⎜ 2 + 2 ⎟⎟ + X
∂x
∂y ⎠
∂y ⎠
⎝ ∂x
⎝ ∂x
⎛ ∂ 2v ∂ 2v ⎞
⎛ ∂v
∂v ⎞
∂p
ρ ⎜⎜ u + v ⎟⎟ = − + μ ⎜⎜ 2 + 2 ⎟⎟ + Y
∂y ⎠
∂y
∂y ⎠
⎝ ∂x
⎝ ∂x
(D.2)
(D.3)
energy equation:
⎛ ∂ 2T ∂ 2T ⎞
⎛ ∂T
∂T ⎞
+v
ρc p ⎜⎜ u
⎟⎟ = k ⎜⎜ 2 + 2 ⎟⎟ + μΦ + q&
∂y ⎠
∂y ⎠
⎝ ∂x
⎝ ∂x
energy transport
thru convection
heat transport thru
conduction
(D.4)
heat generation
⎧⎪⎛ ∂u ∂v ⎞2 ⎡⎛ ∂u ⎞2 ⎛ ∂v ⎞2 ⎤⎫⎪
where μΦ = μ ⎨⎜ + ⎟ + 2 ⎢⎜ ⎟ + ⎜ ⎟ ⎥⎬
⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦⎪⎭
⎪⎩⎝ ∂y ∂x ⎠
(D.5)
viscous dissipation
„
species equations:
⎛ ∂ 2C A ∂ 2C A ⎞
∂C A
∂C A
⎟ + N& A
u
+v
= DAB ⎜⎜
+
2
2 ⎟
∂x
∂y
∂
∂
x
y
⎝
⎠
(D.6)
Boundary layer approximations:
u >> v
⎤
∂u
∂u ∂v ∂v ⎥⎥ Velocity boundary layer
,
,
>>
∂y
∂x ∂y ∂x ⎥⎦
∂T
∂T ⎤
>>
Thermal boundary layer
⎥
∂y
∂x ⎦
∂C A
∂C A ⎤
>>
Concentration boundary layer
⎥
∂y
∂x ⎦
„
Usual additional simplifications: incompressible,
constant properties, negligible body forces, nonreacting,
no energy generation
Eqs. 6.27 is unchanged and the x-mom equation reduces to
∂u
∂u
1 dp∞
∂ 2u
u
+v
=−
+ν 2
ρ dx
∂x
∂y
∂y
(6.28)
The energy equation reduces to
∂T
∂T
∂ T ν ⎛ ∂u ⎞
+v
= α 2 + ⎜⎜ ⎟⎟
∂x
∂y
c p ⎝ ∂y ⎠
∂y
2
u
2 usually negligible
(6.29)
energy transport heat transport thru
thru convection conduction
and the species equation becomes
∂C A
∂C A
∂ 2C A
u
+v
= DAB
∂x
∂y
∂y 2
„
(6.30)
For incompressible, constant property flow, Eq. (6.28) is
uncoupled from (6.29) & (6.30). Eqs. (6.27) & (6.28) need to be
solved first, for the velocity field u(x, y) and v(x, y), before (6.29)
& (6.30) can be solved, i.e., the temperature and species fields are
coupled to the velocity field.
6.5 Boundary Layer Similarity: The Normalized
Convection Transfer Equations
6.5.1 Boundary Layer Similarity Parameters
y
x
Defining x* ≡
and y* ≡ L L is the characteristic length
L
u and v* ≡ v V is the free stream velocity
u* ≡ V
V
T − Ts
CA − CA,s
T* ≡ T − T and CA * ≡
CA,∞ − CA,s
∞
s
we arrive at the convection transfer equations and their boundary
conditions in nondimensional form, as shown in Table 6.1.
Dimensionless groups:
„ Reynolds no.: ReL ≡ VL
v
„ Prandtl no.:
Pr ≡ v
α
„ Schmidt no.: Sc ≡ v
DAB
The final dimensionless governing eqs.: (6.35)-(6.37)--similar in form
∂ u * ∂v *
+
=0
∂ x * ∂y *
2
dp *
∂u *
∂u *
∂
u*
1
(6.35)
u*
+ v*
=−
+
2
∂x *
∂y *
dx * Re L ∂ y *
∂T *
∂T *
∂ 2T *
1
u*
+ v*
=
(6.36)
∂x *
∂y * Re L Pr ∂ y *2
∂ CA *
∂C A *
∂ 2CA *
1
u*
+v*
= Re Sc
∂x *
∂y *
L
∂ y *2
(6.37)
6.5.2 Functional Form of the Solutions
From (6.35), we can write
dp * ⎞
⎛
u* = f ⎜ x*, y*, ReL ,
⎟
dx * ⎠
⎝
(6.44)
where dp*/dx* depends on the surface geometry.
The shear stress and the friction coefficient at the surface
are
⎛ μV ⎞ ∂ u *
∂u
τs = μ
=⎜ L ⎟
∂y y=0 ⎝ ⎠ ∂y * y*= 0
τs
2
∂u *
2
Cf =
=
=
f ( x*, ReL )
2
Re
ρV / 2
L ∂y *
ReL
y* =0
(6.45, 46)
For a prescribed geometry, (6.45) is universally applicable.
From (6.36)
dp * ⎞
⎛
T * = f ⎜ x*, y*, ReL , Pr ,
⎟
dx * ⎠
⎝
k f (T∞ − Ts ) ∂ T *
k f ∂T *
h = − L (T − T )
=+ L
s
∞ ∂ y * y*=0
∂y * y*=0
Nusselt number can be defined as
∂T *
hL
Nu ≡ k = +
f
∂ y * y*=0
For a prescribed geometry, from (6.47)
Nu = f ( x*, ReL , Pr )
The spatially average Nusselt number is then
hL
Nu =
= f (ReL , Pr )
kf
hm L
Similarly for mass transfer,
Sh =
= f (ReL , Sc )
DAB
EX 6.5
(6.47)
(6.48)
(6.49)
(6.50)
(6.54)
6.6 Physical Significance of the Dimensionless
Parameters
See Table 6.2.
„ For heat transfer, the important dimensionless
parameters are:
Re, Nu, Pr, Bi, Fo, Pe (= RePr), Gr
„ For mass transfer,
Re, Sh, Sc, Bim, Fom, Le (= α/DAB)
For boundary layer thicknesses,
δt
δ
δ
1
≈ Pr n ,
≈ Sc n ,
≈ Le n n = for laminar boundary layer
δt
δc
δc
3
(6.55, 56, 58)
δ ≈ δ t ≈ δ c for turbulent boundary layer (why?)
6.7 Boundary Layer Analogies
6.7.1 The Heat and Mass Transfer Analogy
„
Since the dimensionless relations that govern the
thermal and the concentration boundary layers are the
same, the heat and mass transfer relations for a
particular geometry are interchangeable.
With
Nu = f ( x*, ReL )Pr n
and
Sh = f ( x*, ReL )Sc n
and equivalent functions, f (x*, ReL),
Then,
Nu
Sh
hL / k hm L / DAB
= n →
=
n
n
Pr
Sc
Pr
Sc n
1−n
h =
k
and we have hm D Le n = ρ c p Le , n=1/3 for most applications
AB
(6.60)
EX 6.6
6.7.3 The Reynolds Analogy
„
For dp*/dx* = 0 and Pr = Sc = 1, Eqs. 6.35-6.37 are of precisely
the same form. Moreover, if dp*/dx*=0, the boundary conditions,
6.38-6.40 also have the same form. From Equations 6.45, 6.48,
6.52, it follows that
ReL
(6.66)
Cf
= Nu = Sh
2
„
„
Replacing Nu and Sh with Stanton number (St) and mass transfer
Stanton number (Stm), as defined in (6.67) and (6.68), we have
C f / 2 = St = Stm -- Reynolds Analogy
(6.69)
h
h
Nu
Sh
=
, Stm ≡ m =
St ≡
ρVc p RePr
V
ReSc
The modified Reynolds Analogy (or Chilton-Colburn Analogy) is
Cf
2
= StPr
2/3
Cf
Sh
Nu
2/3
=
St
Sc
≡
j
=
;
≡ jH =
m
m
RePr1/ 3 2
ReSc1/ 3
(6.70,71)
Eqs. 6.70, 71 are appropriate for laminar flow when dp*/dx* ~ 0; for
turbulent flow, conditions are less sensitive to pressure gradient.
6.7.2 Evaporative Cooling
For evaporative cooling shown in Fig. 6.10,
"
qconv
+ q"add = q"evap
where "
qevap = nA" h fg (6.62)
"
If qadd is absent, then
h(T ∞ − T s ) = h fg hm [ ρA,sat (T s ) − ρA,∞ ]
⎛ hm ⎞
→ T∞ − Ts = h fg ⎜ ⎟ [ ρ A,sat (Ts ) − ρ A,∞ ]
(6.64)
h
⎝ ⎠
Using the heat and mass transfer analogy, (6.64) becomes
M A h fg
pA,sat (T s ) pA,∞
(T∞ − T s ) =
− T ]
(6.65)
2 /3 [
T
ℜρ c p Le
s
∞
Note: Gas properties ρ, cp, Le should be evaluated at the arithmetic
mean temperature of the thermal b. l., Tam= (Ts+T∞)/2.
EX 6.7
Special Topic—Heat Pipes
Representative Ranges of Convection Thermal Resistance
h (W/m2K)
Areal Rth,conv (Kcm2/W)
2~25
20~200
100~1,000
5,000~400
500~50
100~10
20~200
200~2,000
1,000~10,000
40,000
2,400~49,300
500~50
50~5
10~1
0.25
4.1~0.20
0.049*
___________________________________________________________________
Natural Convection
Air
Oils
Water
Forced Convection
Air
Oils
Water
Microchannel Cooling
Impinging Jet Cooling
Heat Pipe
_________________________________________________________________________
* based on THERMACORE 5 mm heat pipe having capacity of 20W with ΔT=5K.
The Working Principle of Heat Pipes
--Based on phase-change heat transfer
„ Evaporation Section
„ Adiabatic Section
„ Condensation Section
The performance of a heat pipe is dictated by its thermal
resistance Rth and maximum heat load, Qmax, which are
mainly determined by the evaporation characteristics.
A small amount of working fluid (mostly water) is filled in
the heat pipe after it is evacuated.
管壁
„container
„Working fluid
„wick
溝槽
Advantages of Heat Pipes
„superior heat spreading ability
„fast thermal response
„light weight and flexible
„low cost
„simple without active units
燒結金屬
金屬網