convection processes of boiling and condensationocw.snu.ac.kr/sites/default/files/note/2252.pdf ·...
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CONVECTION PROCESSES OF BOILING AND CONDENSATION
• Dimensionless Parameters
• BoilingPool Boiling Forced Convection Boiling
• CondensationLaminar Film CondensationTurbulent Film Condensation
Dimensionless ParametersTsat
g
Ts
ΔT = Ts – Tsat
g(ρl – ρv), hfg,σ, L,ρ, cp, k,μΔT,h = h[ ]
L
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=⎜ ⎟
⎝ ⎠
Boiling• Pool boiling
• Forced convection boiling
solid
liquid
liquid
• Subcooled boilingTemperature of the liquid: below the saturation temperature
Bubbles formed at the solid surface: condense in the liquid
• Saturated boiling
Tsat = Tsat(p)
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
Pool Boiling• Saturated pool boiling
Nukiyama’s power-controlled heating apparatus: boiling curve
power setting (or ): independent variablewire temperature (or ΔTe): dependent variable
sq′′
Nukiyama’s boiling curve
Nichrome: Tm = 1500 KPlatinum: Tm = 2045 K
Modes of pool boiling
nucleate transition filmfreeconvection
oscillation between three allowed values subjected to some small external disturbances
• 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 ⋅
• 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.
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:
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)
• 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)
• 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)
• 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′ = + −
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 −
=−
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′′
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′′ ′′= = =
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 = = =
Example 10.2
Find: Power dissipation per unit length for the cylinder, sq′
Assumption:Steady-state, atmospheric pressure, water at uniform temperature Tsat = 100C,
: 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
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 =′
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
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:
• Two-phase flowvertical tube subjected to a constant heat flux
slug
liquid forced convection
subcooledflow boiling
bubbly
mist
vapor forced convection
annular
transition
h
saturated vapor
saturated liquid
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′′ =
( , )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′′
=
( )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
Condensation
film Dropwisecondensation
Homogeneouscondensation
Direct contactcondensation
• 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
• Laminar film condensation on a vertical plate
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=
⎞∂=⎟∂ ⎠
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
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−
=
( )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⎛ ⎞′′ = = =⎜ ⎟⎝ ⎠
( ) ( )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
=
( )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
⎡ ⎤−= ⎢ ⎥
−⎢ ⎥⎣ ⎦
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>>
• Turbulent film condensation
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−
=
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
⎡ ⎤−⎢ ⎥=⎢ ⎥′⎣ ⎦
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
⎡ ⎤−⎢ ⎥= − +⎢ ⎥′⎣ ⎦
Modified Nusselt number for condensation on a vertical plate
• 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 −=
• 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′ ≡ + −
• 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= ≤