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= ≤