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Transcript of Chapter 6 Fundamental Concepts of Convectionocw.nthu.edu.tw/ocw/upload/77/855/HTchap6.pdf · 6.5.2...
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Chapter 6
Fundamental Concepts of Convection
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6.1 The Convection Boundary LayersVelocity boundary layer:surface shear stress: local friction coeff.:
Thermal boundary layer:local heat flux:
local heat transfer coeff.:
s = uy y=0Cf
su
2 / 2
q s = kf Ty y=0 "0
/f yss s
k T yqhT T T T
=
= =
(6.2)
(6.1)
(6.3)
(6.5)
-
Concentration boundary layer:local species flux:
or (mass basis):
local mass transfer coeff.:
or (mass basis):
"0,
, , , ,
/AB A yA sm
A s A A s A
D C yNh
C C C C=
= =
hm=DAB A / y y=0
A,s A,
(6.6)
(6.9)
" 2
0
[kmol/s m ]AA ABy
CN Dy =
=
" 2
0
[kg/s m ]AA ABy
n Dy
=
=
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6.1.4 Significance of the Boundary LayersVelocity boundary layer: always exists for flow over any surfaceThermal boundary layer: exists if the surface and free stream temperature differConcentration boundary layer: exists if the surface concentration of a species differs from the free stream value
The principal manifestations and boundary layer parameters areVelocity boundary layer: surface friction and friction coefficient CfThermal boundary layer: convection heat transfer and heat transfer convection coefficient hConcentration boundary layer: convection mass transfer and mass transfer convection coefficient hm
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6.2 Local and Average Convection Coefficients6.2.1 Heat TransferThe 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, if Ts is uniform
where .
( ),s shA T T=
( )s s
s s sA Aq q dA T T hdA= =
1s
sAs
h hdAA
=
(6.10-12)
(6.13)
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6.2.2 Mass TransferSimilar for mass transfer of species A, we have
--local--average
orwhere
sm m sA
h h dA= --Mass transfer rate
--Molar transfer rate
" 2A A, A,
A A, A,
( ) [kg/s m ]
( ) [kg/s]m s
m s s
n h
n h A
=
=
(6.15)
(6.18)
(6.16)
(6.19)
" 2A A, A,
A A, A,
( ) [kmol/s m ]
( ) [kmol/s]m s
m s s
N h C C
N h A C C
=
=
-
We can also write Ficks law on a mass basis by multiplying Eq. 6.7 by MA to yield
The value of CA,s or A,s can be determined by assuming thermodynamic equilibrium at the interface between the gas and the liquid or solid surface. Thus,
" AA AB A, A,
0
AB A 0
A, A,
( )
/
m sy
ym
s
n D hy
D yh
=
=
= =
=
satA,
( )ss
p TCT
=R
(6.20)
(6.21)
(6.22)
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6.2.3 The Problem of ConvectionThe 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 ) is viewed as the problem of convection.
However, the problem is not a simple one, as
EXs 6.1-6.3
or mh h
,
or (fluid properties, surface geometry, flow conditions)
i.e., , , ,
m
f p f
h h f
k c
=
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6.3 Laminar and Turbulent FlowCritical Reynolds no. where transition from laminar boundary layer
to turbulent occurs:
EX 6.4
Rex,c =uxc
= 5 105
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6.4 The Boundary Layer Equations (2-D, Steady)
6.4.1 Boundary Layer Equations for Laminar FlowFor the steady, two-dimensional flow of an incompressible
fluid with constant properties, the following equations are involved. (Read 6S.1 for detailed derivation.)continuity eq.: 0=
+
yv
xu
(D.1)
-
momentum eq. (incompressible):
energy equation:
species equations:
Xyu
xu
xp
yuv
xuu +
+
+
=
+
2
2
2
2
Yy
vx
vyp
yvv
xvu +
+
+
=
+
2
2
2
2y-dir.
x-dir.
A2
2
2
2
AB NyC
xCD
yCv
xCu AAAA &+
+
=
+
(D.6)
(D.2)
(D.3)
qyT
xTk
yTv
xTucp &++
+
=
+ 2
2
2
2
2 22
where 2u v u vy x x y
= + + +
(D.4)
(D.5)
energy transport thru convection
heat transport thru conduction
heat generation
viscous dissipation
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Boundary layer approximations:
Usual additional simplifications: incompressible, constant properties, negligible body forces, nonreacting, no energy generation
Thermal boundary layerT Ty x
>>
Velocity boundary layer, ,
u vu u v vy x y x
>> >>
Concentration boundary layerA AC Cy x
>>
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Eqs. 6.27 is unchanged and the x-mom equation reduces to
The energy equation reduces to
and the species equation becomes
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.
2
2
AByCD
yCv
xCu AAA
=
+
(6.30)
(6.29)2
2
2
+
=
+
yu
cyT
yTv
xTu
p
usually negligible
energy transport thru convection
heat transport thru conduction
2
2
1 dpu u uu vx y dx y
+ = + (6.28)
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6.5 Boundary Layer Similarity: The Normalized Convection Transfer Equations
6.5.1 Boundary Layer Similarity ParametersDefining and L is the characteristic length
and V is the free stream velocity
and
we arrive at the convection transfer equations and their boundary conditions in nondimensional form, as shown in Table 6.1.
Dimensionless groups:Reynolds no.: Prandtl no.:Schmidt no.:
LVLRe v
x* xL y* yL
u* uV v* vV
T* T TsT Ts
CA * CA CA,s
CA, CA,s
vPr
AB
vSc D
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The final dimensionless governing eqs.: (6.35)-(6.37)--similar in formu *x *
+v *y *
= 0
u* u *x*
+ v* u*y*
= dp*dx *
+ 1Re L 2u *y*2
u* T *x *
+ v* T *y*
= 1ReL Pr 2T *y*2
u* CA *x *
+ v* CA *y*
= 1ReL Sc 2CA *y*2
(6.35)
(6.37)
(6.36)
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6.5.2 Functional Form of the SolutionsFrom (6.35), we can write
where dp*/dx* depends on the surface geometry.The shear stress and the friction coefficient at the surface
are
For a prescribed geometry, (6.45) is universally applicable.
=
**,*,*,*
dxdpReyxfu L
s = uy y=0
=VL
u*y* y*= 0
C f = s
V 2 / 2= 2Re L
u *y * y*=0
)*,(2 LL
RexfRe
= (6.45, 46)
(6.44)
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From (6.36)
Nusselt number can be defined as
For a prescribed geometry, from (6.47)
The spatially average Nusselt number is then
Similarly for mass transfer,
EX 6.5
=
**,,*,*,*
dxdpPrReyxfT L
h = k fL
(T Ts )(Ts T )
T *y* y*=0
= +k fL
T *y * y*=0
Nu hLk f= +
T *y* y*=0
( )PrRexfNu L ,*,=
( )PrRefk
LhNu Lf
,==
( )ScRefD
LhSh Lm ,AB
==
(6.50)
(6.48)
(6.47)
(6.49)
(6.54)
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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), GrFor mass transfer,Re, Sh, Sc, Bim, Fom, Le (= /DAB)
For boundary layer thicknesses,1, , for laminar boundary layer3
for turbulent boundary layer (why?)
n n nt
t c c
t c
Pr Sc Le n
=
(6.55, 56, 58)
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6.7 Boundary Layer Analogies6.7.1 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
and
and equivalent functions, f (x*, ReL),
Then,
and we have , n=1/3 for most applications
EX 6.6
( ) nL PrRexfNu *,=
( ) nL ScRexfSh *,=
// m ABn n n n
h L DNu Sh hL kPr Sc Pr Sc
= =
hhm
= kDAB Le
n = cpLe1n
(6.60)
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6.7.3 The Reynolds AnalogyFor 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
Replacing Nu and Sh with Stanton number (St) and mass transfer Stanton number (Stm), as defined in (6.67) and (6.68), we have
The modified Reynolds Analogy (or Chilton-Colburn Analogy) is
Eqs. 6.70, 71 are appropriate for laminar flow when dp*/dx* ~ 0; for turbulent flow, conditions are less sensitive to pressure gradient.
, mmp
hh Nu ShSt StVc RePr V ReSc
= =
ShNuReC Lf ==2
/ 2f mC St St= = -- Reynolds Analogy
2/31/32
fm m
C ShSt Sc jReSc
= =2/3 1/3 ;2f
H
C NuStPr jRePr
= =
(6.69)
(6.70,71)
(6.66)
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6.7.2 Evaporative CoolingFor evaporative cooling shown in Fig. 6.10,
where
If is absent, then
Using the heat and mass transfer analogy, (6.64) becomes
Note: Gas properties , cp, Le should be evaluated at the arithmetic mean temperature of the thermal b. l., Tam= (Ts+T)/2.
qconv" + qadd
" = qevap"
qevap" = nA
" hfg
h(T Ts ) = hfghm[A,sat (Ts ) A, ]
A,sat A,[ ( ) ]ms fg shT T h Th
=
(T Ts ) =MAhfg
cpLe2/3 [
pA,sat (Ts )Ts
pA,T
]
qadd"
EX 6.7
(6.65)
(6.64)
(6.62)
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h (W/m2K) Areal Rth,conv (Kcm2/W)___________________________________________________________________Natural Convection
Air 2~25 5,000~400Oils 20~200 500~50Water 100~1,000 100~10
Forced ConvectionAir 20~200 500~50Oils 200~2,000 50~5Water 1,000~10,000 10~1
Microchannel Cooling 40,000 0.25Impinging Jet Cooling 2,400~49,300 4.1~0.20Heat Pipe 0.049*_________________________________________________________________________
* based on THERMACORE 5 mm heat pipe having capacity of 20W with T=5K.
Representative Ranges of Convection Thermal ResistanceSpecial TopicHeat Pipes
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The Working Principle of Heat Pipes
Evaporation SectionAdiabatic SectionCondensation 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.
--Based on phase-change heat transfer
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containerWorking fluidwick
Advantages of Heat Pipessuperior heat spreading abilityfast thermal responselight weight and flexiblelow costsimple without active units
A small amount of working fluid (mostly water) is filled in the heat pipe after it is evacuated.
The Working Principle of Heat Pipes