Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

1

Boiling and Condensation

Boiling = Evaporation at a solid-liquid interface

( ) esats ThTThq Δ⋅=−⋅=ʹ′ʹ′ ΔTe = Excess Temperature

Modes of Boiling:

defined by the kind of flow: - Pool boiling (natural/free convection) - Boiling with forced convection (e.g. pipe flow)

defined by the temperature range: - Saturated boiling: Tfluid = Tsat - Subcooled boiling: Tfluid < Tsat

Chap. 16: Convection with Phase Change

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

2

Pool Boiling – Boiling Curve

Experiment of Nukiyama:

WAqqVIq =ʹ′ʹ′⋅= I

VRRfT WWW == )(

( )satW TTqh−

ʹ′ʹ′=

h is determined by electric properties I, V and the wire surface AW

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

3

Regions of the Boiling Curve

C: maximum heat flux, end of nucleate boiling

D: Leidenfrost point, start of film boiling

P: maximum h A

B

P

C

D

103

104

105

106

1000 120 30 10 5 1

)( CTTT satse °−=Δ

KmWq⋅

ʹ′ʹ′2

A: Start nucleate boiling

B: Start nucleate jet boiling

Chap. 16.1: Pool Boiling

]/[ 2mWq ʹ′ʹ′

Nucleate boiling Film boiling Transition

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

4

Example: Boiling of methanol in a horizontal tube

Photographien von Prof. J.W. Westwater, University of Illinois at Champaign-Urbana

1. Nucleate boiling (jets and columns)

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

5

2. Transition boiling

3. Film boiling

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

6

Free convection boiling (ΔTe < 5 K)

Before the onset of nucleation, heat transfer only due to (natural) convection

Temperature distribtion inside of liquid depending on height z

Top surface is superheated (T0 - Tsat)

( )

( )34

45

:

:

Tqturbulent

Tqlaminar

Δ∝ʹ′ʹ′

Δ∝ʹ′ʹ′

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

7

Formation of Bubbles

γπππ ⋅⋅+⋅⋅=⋅⋅ RPRPR lb 222

Surface of the vapor bubble has a surface tension: equilibrium of forces:

Pressure inside

Pressure outside

Surface tension = +

RPPP lb

γ⋅=−=Δ2

To overcome the surface tension, there has to be an excess pressure inside the bubble:

Excess pressure requires an excess temperature of the liquid

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

8

3,

21

Pr)(

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅

Δ⋅⎥⎦

⎤⎢⎣

⎡ −⋅⋅⋅=ʹ′ʹ′ n

llv

lpvllvls hC

Tcghqγ

ρρµ

µl Viscosity of liquid hlv Evaporation enthalpy ρ Density γ Surface tension l, v Indices for liquid and vapor C, n experimental constants

Nucleate Pool Boiling

Empirical: Interaction between surface properties and bubble formation in the liquid

Correlation by Rohsenov:

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

9

Combination Fluid/ Surface

C

n

Water/Copper scored polished

0.0068 0.0130

1.0 1.0

Water/Steel polished

0.0130

1.0

Water/Nickel 0.0060 1.0

Water/Platinum 0.0130 1.0

n-Pentane/Copper polished

0.0154

1.7

Constants for the correlation of Rohsenov

C = Interation between fluid and surface

n = Fluid property

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

10

41

2max)(⎥⎦

⎤⎢⎣

⎡ −⋅⋅⋅⋅=ʹ′ʹ′

v

vlvlv

ghCqρ

ρργρ

41

2min )()(⎥⎦

⎤⎢⎣

⎡

+

−⋅⋅⋅⋅⋅=ʹ′ʹ′

vl

vlvlv

ghCqρρρργ

ρ

Critical / Maximum Heat Flux

If heat flux is higher: transition to film boiling (ΔT > 1000 K)

Minimal Heat Flux (at Leidenfrost point)

If heat flux is smaller: collapse of film boiling

C = 0.149 for large horizontal plates

C = 0.09 for large horizontal plates

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

11

41

3

)()(

⎥⎦

⎤⎢⎣

⎡

−⋅⋅

⋅ʹ′⋅−⋅⋅=

⋅=

satsvv

lvvl

v

convD TTk

DhgCkDhuN

νρρ

( )satsvplvlv TTchh −⋅⋅+=ʹ′ ,80.0

Film Boiling

Nusselt correlation for cylinder or sphere with diameter D

Cylinder: C = 0.62 Sphere: C = 0.67

Correction for latent heat:

For Ts > 300°C: thermal radiation has to be considered as well

Chap. 16.1: Pool Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

12

Boiling in a vertical pipe: Two-Phase Flow

Chap. 16.2: Forced Convection Boiling

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

13

Heat Transfer with Condensation

Occurs when Tw < Tsat

Modes of condensation

Direct condensation:

Spray of vapor entering liquid

Homogeneous condensation:

Mixture of hot humid gas with cold gas, formation of fog

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

14

Laminar Film Condensation on a Vertical Plate

Assumptions (Analysis by Nusselt):

•  laminar film flow, constant properties of fluid

•  Gas phase is pure vapor at Tsat, no heat conduction in the vapor

•  no friction between vapor and liquid, i.e.

•  no thermal boundary layer inside the vapor

•  no convective transport (heat and momentum) inside the liquid boundary layer

0=∂

∂

=δyyu

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

15

System to be considered:

- Velocity profile without gradient at outer surface

- Temperature profile inside boundary layer is linear

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

16

g

yu

dxdp

yuv

xuu l

g

negligibleconvection

l

v

⋅−⋅+−=⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅=

ρ∂∂

µ∂∂

∂∂

ρ

ρ

2

2

)(0

2

2

)(0

yT

yuv

xuuc f

negligibleconvection

p ∂∂

α∂∂

∂∂

ρ ⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⋅

=

Mathematical model for liquid film:

Mx

E

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

17

)(2

2

vll

gyu

ρρµ∂

∂−⋅−=

Momentum equation:

With boundary conditions:

0:0:0 =∂

∂===

yuyuy δ

Solution:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛⋅−⋅⋅−⋅

=22

21)()(

δδµδρρ yygyu

l

vl

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

18

( )l

vllx

fgxdyyu

bxm

µδρρρ

ρδ

⋅

⋅−⋅⋅=Γ=⋅⋅= ∫ 3

)()()( 3)(

0

Calculation of mass flux in the film (per unit width):

onCondensati

lv

Conduction

s mdhdxbyTkdxbq ⋅=⋅⋅∂

∂⋅−=⋅⋅ʹ′ʹ′

δ(x) is unknown, can be determined using the energy balance:

Calculate temperature profile:

02

2=

yT

∂

∂sat

s

TTy

TTy

==

==

)(:

)0(:0

δδE Boundary conditions:

dxdhh

dxmd

bq lvlvs

Γ⋅=⋅⋅=ʹ′ʹ′

12

1

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

19

sssat TyTTT +⋅⎟⎠

⎞⎜⎝

⎛ −=

δSolution: linear profile

δssat

ly

lsTTk

dydTkq −

⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=ʹ′ʹ′

=0

Heat flux:

( )lv

ssatl

hTTk

dxd

⋅

−⋅=

Γ

δSubstituting in 2

Substituting with 1

( )( )

dxhgTTkd

lvvll

ssatll ⋅⋅−⋅⋅

−⋅⋅=⋅

ρρρµ

δδ 3

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

20

( )( )

414)( ⎥

⎦

⎤⎢⎣

⎡⋅

⋅−⋅⋅

−⋅⋅⋅= x

hgTTkxlvvll

ssatll

ρρρµ

δ

Integration leads to solution for δ(x):

With convective heat transfer inside the film is included, we can use an effective latent heat hlv‘

( )( )lv

ssatlplvlv h

TTcJaNumberJakobJahh

−⋅=⋅+⋅=ʹ′ ,:68.01

)()( 0

xk

TTdydTk

xh l

sats

yl

δ=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−

= =( )( )

413

4)( ⎥

⎦

⎤⎢⎣

⎡

⋅−⋅⋅

ʹ′⋅⋅−⋅⋅=

xTThkgxh

ssatl

lvlvll

µρρρ

Calculation of h value using Fourier‘s Law:

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

21

( )( ) Hx

ssatl

lvlvllH

hHTThkgdxxh

Hh

=⋅=⎥

⎦

⎤⎢⎣

⎡

⋅−⋅

ʹ′⋅⋅−⋅⋅⋅=⋅= ∫ 3

4943.0)(1413

0 µρρρ

( )( )

413

943.0 ⎥⎦

⎤⎢⎣

⎡

−⋅⋅

⋅ʹ′⋅−⋅⋅⋅=

⋅=

ssatll

lvvll

lH

TTkHhg

kHhNu

µρρρ

( )ssat TTAhq −⋅⋅= ( )lv

ssat

lv hTTAh

hqm

ʹ′−⋅⋅

=ʹ′

=

Averaged h value:

Nusselt correlation for laminar film condensation on a vertical plate with height H

Calculation of heat transfer rate and mass transfer rate

Chap. 16.3: Condensation

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

22