convection processes of boiling and condensation · 2018. 1. 30. · • film pool boiling vapor...
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CONVECTION PROCESSES OF BOILING AND CONDENSATION
• Dimensionless Parameters
• BoilingPool Boiling Forced Convection Boiling
• CondensationLaminar Film CondensationTurbulent Film Condensation
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Dimensionless ParametersTsat
g
Ts
ΔT = Ts – Tsat
g(ρl – ρv), hfg,σ, L,ρ, cp, k,μΔT,h = h[ ]
L
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Dimensionless Parametersg(ρl – ρv), hfg,σ, L,ρ, cp, k,μΔT,h = h[ ]
( )l vg2
ρ ρ ρμ−
=
( )l vL
ghLNu fk 2 ,Ja, Pr, Bo
ρ ρ ρμ
⎛ ⎞−= = ⎜ ⎟
⎝ ⎠
p
fg
c Th
JaΔ
= =sensible heatlatent heat
buoyancy forceviscous force
pck
Prμ
=να
= =viscous diffusionthermal diffusion
( )l vg L2
Boρ ρ
σ−
= =gravity forcesurface force
:Jacob No.
:Bond No.
3
2Gr g TLβν
⎛ ⎞Δ=⎜ ⎟
⎝ ⎠
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Boiling• Pool boiling
• Forced convection boiling
solid
liquid
liquid
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• Subcooled boilingTemperature of the liquid: below the saturation temperature
Bubbles formed at the solid surface: condense in the liquid
• Saturated boiling
Tsat = Tsat(p)
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Newton’s law of cooling
( )s sq h T Tsat′′ = − eh TΔ=
ΔTe: excess temperatureformation of vapor bubbledetach process from the surface
• Vapor bubble growth and dynamicsexcess temperaturenature of surfacethermopysical properties
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Pool Boiling• Saturated pool boiling
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Nukiyama’s power-controlled heating apparatus: boiling curve
power setting (or ): independent variablewire temperature (or ΔTe): dependent variable
sq′′
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Nukiyama’s boiling curve
Nichrome: Tm = 1500 KPlatinum: Tm = 2045 K
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Modes of pool boiling
nucleate transition filmfreeconvection
oscillation between three allowed values subjected to some small external disturbances
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• Free convection boiling
• Nucleate boilingisolated bubbles: heat transfer dominant by
fluid mixing near surfacejets or columns: small values of the excess
temperature change cause high rates of heat transfer
( )1/ 4~ ,eh TΔ
( )4 / 3~s eq TΔ′′( )1/ 3~ ,eh TΔ
( )5 / 4~s eq TΔ′′laminar:turbulent:
4 2~ 10 W/m Kh ⋅
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• Transition boiling(unstable film boiling, partial film boiling)
blanket begins to form on the surfaceoscillation between film and nucleate boiling
eTΔ∝film surface increase in total surface
• Film boilingLeidenfrost point: completely covered by a
vapor blanketHeat transfer by conduction only through the
vapor film, radiation becomes dominant.
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Pool Boiling Correlations• Nucleate pool boiling
number of surface nucleate sitesthe rate at which bubble originate from each site
Nu Re Prfc fcm nL fc LC=
( )bl v
Dg
σρ ρ
∝−
From force balance between buoyancy and surface tension
characteristic length scale:
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characteristic velocity scale:
tb: the time between bubble departures
( )3 2/b b s
b l fgl fg b s b
D D qV
t hh D q D ρρ
′′∝ ∝ ∝
′′
correlation for nucleate boiling
( ) 31/ 2
,
, Prp l el v
s l fg ns f fg l
c Tgq h
C hρ ρ
μσ
⎛ ⎞⎡ ⎤ Δ−′′ = ⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎝ ⎠
for Cs,f and n: see Table 10.1
Rohsennow (1952)
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• Critical heat flux for nucleate boilingOperation of a boiling process:
close to the critical pointdanger of dissipating heat in excess
large horizontal cylinder, sphere, large finitesurfaces: within 16% deviation 24/ 0.131C π= ≈
large horizontal plates: C = 0.149
( ) 1/ 4
max 2l v
fg vv
gq Ch
σ ρ ρρ
ρ⎡ ⎤−
′′ = ⎢ ⎥⎣ ⎦
properties at saturation temperature
Zuber (1958)
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• Minimum heat flux
for large horizontal plates
( )( )
1/ 4
min 2l v
v fg
l v
gq C h
σ ρ ρρ
ρ ρ
⎡ ⎤−′′ ⎢ ⎥=
+⎢ ⎥⎣ ⎦
properties at saturation temperature
C = 0.09 accurate to about 50%
Zuber (1958)
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• Film pool boilingvapor film blanket: no contact between the
liquid phase and the surface
C = 0.62 for horizontal cylindersC = 0.67 for spheres
for film boiling on a cylinder or sphere of diameter D
( )( )
1/ 43conv
sat
Nu l v fgD
v v v s
g h Dh DC
k k T Tρ ρ
ν
′⎡ ⎤−= = ⎢ ⎥
−⎢ ⎥⎣ ⎦
properties at film temperature: ( )sat / 2f sT T T= +
( ), sat0.80fg fg p v sh h c T T′ = + −
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at elevated surface temperatures: o300 CsT ≥
significant radiation across the vapor film
4 / 3 4 / 3 1/ 3conv radh h h= +
Bromley (1950)
When rad conv ,h h< conv rad34
h h h= +
( )4 4sat
radsat
s
s
T Th
T Tεσ −
=−
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Example 10.1
Assumption:Steady-state, atmospheric pressure, water at uniform temperature Tsat = 100ºC, large pan bottom: polished copper, negligible heat loss from heater to surroundings
Find: 1) The power required to boil water in the pan, 2) Evaporation rate due to boiling, 3) Estimation of critical heat flux,
bmsq
maxq′′
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1) Power: nucleate boiling
( ) 31/ 2,
, Prp l el v
l fg ns f fg l
s
c Tq
gh
C hρ ρ
μσ
⎛ ⎞⎡ ⎤ Δ−= ⎜ ⎟⎢ ⎥ ⎜
⎝′ ⎟⎣ ⎦ ⎠′
properties:6 2
3
3
3
,
279 10 N s/m2257 kJ/kg
957.9 kg/m
0.5955 kg/m
58.9 10 N/m4.217 kJ/kg K
Pr 1.76
l
fg
l
v
p l
l
h
c
μ
ρ
ρ
σ
−
−
= × ⋅
=
=
=
= ×= ⋅
=
118 100 18 CeTΔ = − =
, 0.0128, 1.0s fC n= =From Table 10.1
2836 kW/m=
2
59.1 kW4s ssq Dq A q π′′ ′′= = =
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2) Evaporation rate
bs fgmq h=
3) Critical heat flux
( ) / 4
2max
1
l vfg v
v
gChq
σ ρ ρρ
ρ⎡ ⎤−
= ⎢ ⎥⎣ ⎦
′′
C = 0.149 for large horizontal plate
x2
ma 1.26 MW/mq′ =′
0.0262 kg/s 94 kg/hs
fgb
qh
m = = =
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Example 10.2
Find: Power dissipation per unit length for the cylinder, sq′
Assumption:Steady-state, atmospheric pressure, water at uniform temperature Tsat = 100C,
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: film boiling255 100 155 CeTΔ = − =
( )( )
1/ 43
sat
convNu ,l v fgD
v v v s
g h Dh DC
k k T Tρ ρ
ν
′⎡ ⎤−= = ⎢ ⎥
−⎢ ⎥⎣ ⎦
( ), sat0.80fg fg p v sh h c T T′ = + −
( )ssq q Dπ′′′ =
4/ 3 4 / 3 1/ 3conv radhh h= +
( )4 4sat
satrad
s
s
T Th
T Tεσ −
=−
( ) eTh Dπ= Δ
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properties:3 3
6 2,
1 / 957.9 kg/m , 2257 kJ/kg, 0.4902 kg/m ,
1.980 kJ/kg K, 0.0299 W/m K, 279 10 N s/ml f fg v
p v v v
v h
c k
ρ ρ
μ −
= = = =
= ⋅ = ⋅ = × ⋅
( )( )
1/ 432
satconv 238 W/m Kl v fgv
v v s
g h DkC
D k Th
Tρ ρ
ν
′⎡ ⎤−= = ⋅⎢ ⎥
−⎢ ⎥⎣ ⎦
( )4 4sat 2
sr
aad
t
21.3 W/m Ks
s
T TT
hT
εσ −= = ⋅
−
4/ 3 4 / 3 1/ 3238 21.3h = +
2254.1 W/m Kh = ⋅
( ) 742 W/mes Tq h Dπ= Δ =′
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Forced Convection Boiling
• External flow• Internal flow: Two-phase flow
• External forced convection boilingeffect of forced convection and subcooling:
increase the critical heat flux
Ex) water at 1 atmpool boiling: 1.3 MW/m2
convection boiling: 35 MW/m2
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for a crossflow over a cylinder of diameter D
low velocity:1/ 3
max 1 41Wev fg D
qh Vρ π
⎡ ⎤′′ ⎛ ⎞⎢ ⎥= + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
V D
high velocity: ( ) ( )3 / 4 1/ 2
max1/ 3
/ /169 19.2 Wel v l v
v fg D
qh V
ρ ρ ρ ρρ π π′′
= +
2
We vD
V Dρσ
= =inertia force
surface tension
high and low velocity region:1/ 2
max 0.275 1l
v fg v
qh V
ρρ π ρ′′ ⎛ ⎞> +⎜ ⎟<
⎝ ⎠
Weber number:
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• Two-phase flowvertical tube subjected to a constant heat flux
slug
liquid forced convection
subcooledflow boiling
bubbly
mist
vapor forced convection
annular
transition
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h
saturated vapor
saturated liquid
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for saturated boiling region in smooth circular tube
( ) ( )0.70.1
0.64 0.80.16,
sp
0.6683 1 (Fr) 1058 1l ss f
v fg
qh X X f X Gh m h
ρρ
⎛ ⎞′′⎛ ⎞= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟′′⎝ ⎠ ⎝ ⎠
( ) ( )0.70.45
0.0.08 0.80.72,
sp
1.136 1 (Fr) 667.2 1l ss f
v fg
qh X X f X Gh m h
ρρ
⎛ ⎞′′⎛ ⎞= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟′′⎝ ⎠ ⎝ ⎠
0 0.8X< ≤choose larger values of h
all properties: at saturation temperature
hsp: associated with liquid forced convection region
( ) ( )( ) ( )
sp1/ 2 2 / 3
/ 8 Re 1000 PrNu
1 12.7 / 8 Pr 1D
Dl
h D fk f
−= =
+ −
/ cm m A′′ =
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( , )c
cA
u r x XdAX
m
ρ≡∫
Gs,f: Surface-liquid combination
X: time average mass fraction of vapor in fluid
Values of Gs,f for various surface-liquid combination
For negligible changes in fluid’s kinetic and potential energy, and negligible work
( ) s
fg
q DxX x
mhπ′′
=
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( )2/Fr lm
gDρ′′
=Froude number:
f(Fr): stratification parameter
f(Fr) = 1: for vertical tubes and for horizontal tubes with Fr 0.04≥
0.3(Fr) 2.63Frf = for horizontal tubes with Fr 0.04≤
Applicable when channel dimension is large relative to bubble diameter
( )/ 1Co2
l v
h
gD
σ ρ ρ−= ≤ Co: Confinement number
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Condensation
film Dropwisecondensation
Homogeneouscondensation
Direct contactcondensation
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• Condensation thickness: thermal resistance between vapor and surface→ horizontal tube bundles preferred
• Dropwise condensation: favorable to heat transfer→ surface coatings to inhibit wetting
• Condensation design: often based on film condensation
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• Laminar film condensation on a vertical plate
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Nusselt (1916)
Assumptions
1) Laminar flow, constant properties
2) Gas: pure vapor and uniform temperature at Tsat
3) Heat transfer only by condensation (neglect conduction)
4) Negligible Momentum and energy transfer by advection (low film flow velocity)
5) 0y
uy δ=
⎞∂=⎟∂ ⎠
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x- momentum equation2
20 l lu dp g
y dxμ ρ∂
= − +∂
viscous force ~ buoyancy force
vdp gdx
ρ=
( )2
2 l vl
u gy
ρ ρμ
∂= − −
∂
(0) 0, 0y
uuy δ=
⎞∂= =⎟∂ ⎠
Momentum and energy conservation
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condensate mass flow rate per unit width
( ) 22
( , ) 2l v
l
g y yu x yρ ρ δ
μ δ δ⎡ ⎤− ⎛ ⎞= −⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦
( )
0
( ) ( ) ( , )x
lm x x u x y dy
bδ
Γ ρ= = ∫
Then,
( ) 3
( )3
l l v
l
gx
ρ ρ ρ δΓ
μ−
=
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( )fg sdq h dm q bdx′′= =
energy conservation
( )satl ss
k T Tq
δ−
′′ =
s
fg
qddx hΓ ′′
→ =
( )satl s
fg
k T Tddx hΓ
δ−
=( ) 3
( )3
l l v
l
gx
ρ ρ ρ δΓ
μ⎡ ⎤−
=⎢ ⎥⎣ ⎦
( ) ( ) 2satl s l l v
fg l
k T T gd ddx h dx
ρ ρ ρ δΓ δδ μ
− −= =
( )( ) m xxb
Γ =
1s fg fg fg
dm d m dq h h hb dx dx b dx
Γ⎛ ⎞′′ = = =⎜ ⎟⎝ ⎠
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( ) ( )2satl l v l s
l fg
g k T Tddx h
ρ ρ ρ δ δμ δ− −
=
( )( )
sat3 l l s
l l v fg
k T Td dx
g hμ
δ δρ ρ ρ
−=
−
( )( )
1/ 4
sat4( ) l l s
l l v fg
k T T xx
g hμ
δρ ρ ρ
⎡ ⎤−= ⎢ ⎥
−⎢ ⎥⎣ ⎦
sensible heat transfer correction
( )1 0.68Ja ,fg fgh h′ = + Ja p
fg
c ThΔ
=
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( )satl sk T Tδ−
=( )sats x sq h T T′′ = − lx
khδ
→ =
( )( )
1/ 43
sat
,4
l l v l fgx
l s
g k hh
T T xρ ρ ρμ
′⎡ ⎤−= ⎢ ⎥−⎣ ⎦ 0
1 43
L
L x Lh h dx hL
= =∫
( )( )
1/ 43
sat
0.943 l l v l fgL
l s
g k hh
T T Lρ ρ ρμ
′⎡ ⎤−= ⎢ ⎥−⎣ ⎦
( )( )
1/ 43
sat
Nu 0.943 l l v fgLL
l l l s
g h Lh Lk k T T
ρ ρ ρμ
′⎡ ⎤−= = ⎢ ⎥−⎣ ⎦
( )( )
1/ 4
sat4( ) l l s
l l v fg
k T T xx
g hμ
δρ ρ ρ
⎡ ⎤−= ⎢ ⎥
−⎢ ⎥⎣ ⎦
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liquid properties at film temperature
hfg: at Tsat
( )( )
1/ 43
sat
Nu 0.943 l l v fgLL
l l l s
g h Lh Lk k T T
ρ ρ ρμ
′⎡ ⎤−= = ⎢ ⎥−⎣ ⎦
sat
2s
fT TT +
=
inclined plate: cosg g θ→
tube: R δ>>
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• Turbulent film condensation
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mass flow per unit depth
( ) ( )l mx u xΓ ρ δ=
4Rel
δΓμ
≡
Wave-free laminar region
( ) 3
( )3
l l v
l
gx
ρ ρ ρ δΓ
μ−
=
( ) 3
2
4Re
3l l v
l
gδ
ρ ρ ρ δμ−
=
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assuming l vρ ρ>>
( )( )
1/ 4
sat4( ) l l s
l l v fg
k T T xx
g hμ
δρ ρ ρ
⎡ ⎤−= ⎢ ⎥
−⎢ ⎥⎣ ⎦
( )( )
1/ 43
sat
0.943 l l v l fgL
l s
g k hh
T T Lρ ρ ρμ
′⎡ ⎤−= ⎢ ⎥−⎣ ⎦
( ) ( )1/ 32
1/ 3/
1.47 Re 30Re L l
l
h gk δδ
ν− ≤=
( )( )
3 / 4
sat1/ 32
Re 3.78/
l s
l fg l
k L T T
h gδ
μ ν
⎡ ⎤−⎢ ⎥=⎢ ⎥′⎣ ⎦
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Laminar wavy region 30 Re 1800δ≤ ≤
( )1/ 32
1.22
/ Re 1.08Re 5.2
L l
l
h gk
δ
δ
ν=
−
( )( )
0.82
sat1 / 32
3.70Re 4.8
/l s
l fg l
k L T T
h gδ
μ ν
⎡ ⎤−⎢ ⎥= +⎢ ⎥′⎣ ⎦
Turbulent region Re 1800δ ≥
( )( )
1/ 32
0.5 0.75
/ Re8750 58Pr Re 253
L l
l
h gk
δ
δ
ν−
=+ −
( )( )
4 / 3
sat 0.5 0.51 / 32
0.069Re Pr 151Pr 253
/l s
l l
l fg l
k L T T
h gδ
μ ν
⎡ ⎤−⎢ ⎥= − +⎢ ⎥′⎣ ⎦
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Modified Nusselt number for condensation on a vertical plate
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• Film condensation on radial system
C = 0.826 for sphereC = 0.729 for tube
( )( )
1/ 43
sat
l l v l fgD
l s
g k hh C
T T Dρ ρ ρμ
′⎡ ⎤−= ⎢ ⎥−⎣ ⎦
a vertical tier of N tubes
( )( )
1/ 43
,sat
0.729 l l v l fgD N
l s
g k hh
N T T Dρ ρ ρμ
′⎡ ⎤−= ⎢ ⎥−⎣ ⎦
1/ 4,D N Dh h N −=
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• Film Condensation in Horizontal Tubes
for low vapor velocities ,,Re 35,000v m v
v iv
u Dρμ
⎛ ⎞= <⎜ ⎟⎝ ⎠
( )( )
1/ 43
sat
0.555 l l v l fgD
l s
g k hh
T T Dρ ρ ρμ
′⎡ ⎤−= ⎢ ⎥−⎣ ⎦
( ), sat38fg fg p l sh h c T T′ ≡ + −
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• Dropwise Condensation
steam condensation on copper surface
dc sat sat51,104 2044 ( C) 22 C 100 Ch T T= + ≤ ≤
dc sat255,510 100 Ch T= ≤