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ΔT e = Excess Temperature defined by the temperature range: - Saturated boiling: T fluid = T sat - Subcooled boiling: T fluid < T sat Modes of Boiling: Boiling = Evaporation at a solid-liquid interface ME 150 – Heat and Mass Transfer Chap. 16: Convection with Phase Change ThTThq Δ ⋅ = − ⋅ = ʹ′ ʹ′ Prof. Nico Hotz 1
Citation preview
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
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