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Evaporation and condensation in non-equilibrium
Henning Struchtrup
J.P. Caputa
University of Victoria, Canada
Signe Kjelstrup & Dick Bedeaux
NTNU Trondheim
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Example: Condensation/evaporation coefficientHertz—Knudsen—Schrage equation
j =2
2−KC1√2π
µKEpsat (Tl)√RTl
−KCpv√RTv
¶
[Marek&Straub, 2001]
j – mass flux in/out of phase interface
KC/E – condensation/evaporation coefficients
Non-equilibrium Thermodynamics [e.g., Kjelstrup & Bedeaux 2010]
Interface conditions (linearized): dimensionless resistivities rαβ⎡⎢⎢⎢⎣psat(Tl)−p√
2πRTl
− psat(Tl)√2πRTl
Tv−TlTl
⎤⎥⎥⎥⎦ =⎡⎢⎢⎢⎣r11 r12
r21 r22
⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣j
qvRTl
⎤⎥⎥⎥⎦
Onsager symmetry:r21 = r12
positive entropy generation:
r11 ≥ 0 , r22 ≥ 0 , r11r22 − r12r21 ≥ 0
Macroscopic thermodynamics cannot predict values for resistivities
=⇒ needs microscopic modelling and/or experiment
Non-equilibrium phase interfaces
• find coefficients for interfacial mass and energy transfer
• for which processes are non-equilibrium effects important?Ward apparatus? Phillips-Onsager cell? Industrial applications?
• is cold to warm distillation possible?
Kinetic theory and evaporation/condensationmass and energy flux: (in x3 = n direction), velocity distribution function f (ci, xi, t)
j =
∞Z−∞
mcnfdc , Q =
∞Z−∞
m
2cnc
2fdc
distribution at liquid-vapor interface:
fint =
⎧⎪⎨⎪⎩f− , c0n ≤ 0
f+ , cn > 0
incident particles: non-equilibrium distribution
f− = fM (p, T, c)
½1 + cn
j
p+2
5
czpRT
µc2
2RT− 52
¶Q
¾emitted/reflected particles: evaporation + reflection of non-condensing particles
f+ = θe (ck) fM (psat (Tl) , Tl, c) +1
|cn|
Zc0n<0
f−Rc (c0k → ck) |c0n| dc0
θe (ck) — evaporation probability, Rc (c0k → ck) — condensation-reflection kernel
Condensation-reflection kernel Rc¡c0k → ck
¢normalization
Zcn>0
R (c0k → ck) dc = 1
reciprocity relation [Kuscer 1971]: guarantees microreversibility
|c0n| f0 (Tl, c0)Rc (c0k → ck) = |cn| f0 (Tl, c)Rc (−ck →−c0k)
reflection probability α (c0k) and condensation probability θc (c0k)
α (c0k) = 1− θc (c0k) =
Zcn>0
Rc (c0k → ck) dc ≤ 1
evaporation coefficient
θe (ck) = 1−
Rc0n<0
|c0n| f 0M,satRc (c0k → ck) dc0
|cn| fM,sat
evaporation probability equals condensation probability [Sharipov 1994]
θe (ck) = 1−Zc0n<0
RRc (−ck → −c0k) dc0 = θc (−ck)
Model for evaporation and condensation [Caputa & HS 2010]
MD simulations: faster particles condense more easily! [Tsuruta et al 1999]
velocity dependent condensation/evaporation coefficient[Tsuruta et al 1999][Bedeaux/Kjelstrup/Rubi 2003]
θc (c0n) = θe (c
0n) = ψ
∙1− ω exp
µ− c02n2RTl
¶¸
Condensation-reflection: Tsuruta coeff. + Maxwell reflection kernel
Rc (c0k → ck) = [1− θc(c
0n)]
"(1− γ) δ (c0k − ck + 2cjnjnk) + γ
[1− θc(cn)] |cn| f0 (c)Rcn>0
[1− θc(cn)] |cn| f0 (c) dc
#
accommodation coefficient γ (specular reflection and/or thermalization)
Macroscopic interface fluxes [Caputa & HS 2010]compute mass and energy flux at interface: use f− and f+"
j0Q0RTl
#=
"R11 R12
R21 R22
#"jQRTl
#with many abbreviations:
j0 = η (Tl, Tl)psat (Tl)√2πRTl
−η (Tl, Tv)p√2πRTv
,Q0RTl
= 2ϕ (Tl, Tl)psat (Tl)√2πRTl
−2ϕ (Tl, Tv)p√2πRTl
rTvTl
η (Tl, T ) = ψ
µ1− Tlω
Tl + T
¶ϕ (Tl, T ) = (1− γ)ψ
Ã1− ω
¡Tl +
12T¢Tl
(Tl + T )2
!+ γ
T
T + Tl− γ [1− ψ (1− ω)]
£1− ψ(1− 3ω
8)¤
1− ψ(1− ω2)
T 2lT (T + Tl)
+ γψ£1− ψ
¡1− 3ω
8
¢+ ω
8
¤1− ψ
¡1− ω
2
¢ TlT + Tl
R11 =2− ψ
2+
ψω
2
T3/2l
¡Tl +
52Tv¢
(Tl + Tv)5/2
, R12 = −3ψω
10
T5/2l
(Tl + Tv)5/2,
R21 = γ(1− ψ)1− ψ
¡1− 3ω
8
¢1− ψ(1− ω
2)+3
4(1− γ)ψω
√TlT
3v
(Tl + Tv)7/2+21
8ψω
T3/2l T 2v
(Tl + Tv)7/2+ γψω
1− ψ¡1− 3ω
8
¢1− ψ
¡1− ω
2
¢ T 5/2l
¡Tl +
72Tv¢
(Tl + Tv)7/2
− γψω
8
1− ψ (1− 3ω)1− ψ(1− ω
2)
T3/2l T 2v
(Tl + Tv)7/2
R22 =
µ1− γ
2− ψ
2(1− γ)
¶− (1− γ)
ψω
10
T3/2l T 2v
(Tl + Tv)7/2+
ψω
20
[2(11γ − 5)(ψ − 1) + (5− 192γ)ψω]£
1− ψ¡1− ω
2
¢¤ T5/2l
(Tl + Tv)7/2
− 7
40ψω
ψω¡1 + 2
7γ¢− 2
¡1 + 5
7γ¢(1− ψ)
1− ψ(1− ω2)]
T5/2l Tv
(Tl + Tv)7/2.
similar result without microreversibility [Bond & HS 2004]
Linearized fluxes/Onsager resistivities [Caputa & HS 2010]expand for small deviations: ∆T = Tl − Tv , ∆p = psat(Tl)− p⎡⎢⎢⎢⎣
psat(Tl)−p√2πRTl
− psat(Tl)√2πRTl
Tv−TlTl
⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎣r11 r12
r21 r22
⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣j
qvRTl
⎤⎥⎥⎥⎦classical kinetic theory: [Cipolla et al. 1974]ω = 0 (impact independent), γ = 1 (full accomm.), condensation coefficient ψ
rkin. theory =
"1ψ − 0.40044 0.126
0.126 0.294
#
our model: ω = 0, γ = 1, ψ
rω=0,γ=1 =
"1ψ −
716
18
18
14
#=
"1ψ − 0.437 0.125
0.125 0.25
#
differences due to neglect of Knudsen layers
Impact dependent resistivities [Caputa & HS 2010]
full thermalization, γ = 1 thermalization/reflection, γ = 0.5
Onsager symmetry: r12 = r21
resistivities depend on coefficients ω, ψ, γ
Question: can rαβ be determined from experiments?
Measurement of rαβ
4 unknown coefficients: need 4 eqs =⇒ data from 2 runs (A, B)⎡⎢⎢⎢⎣psat
³T(A)l
´Z(A)R
q2πRT
(A)l
F(A)j
psat
³T(A)l
´T(A)l
Z(A)q2πRT
(A)l
F(A)q
⎤⎥⎥⎥⎦ ="r11 r12
r21 r22
#⎡⎢⎣ j(A)q(A)v
RTl
⎤⎥⎦⎡⎢⎢⎢⎣
psat
³T(B)l
´Z(B)R
q2πRT
(B)l
F(B)j
psat
³T(B)l
´T(B)l
ZBq2πRT
(B)l
F(B)q
⎤⎥⎥⎥⎦ ="r11 r12
r21 r22
#⎡⎢⎣ j(B)q(B)v
RTl
⎤⎥⎦Z — real gas correction, Fj ∼ ∆p, Fq ∼ ∆T — thermodynamic forces
MD simulations: [Xu et al.] tabulated all data required, all possible pairs used
0 20 40 60 800.0
0.2
0.4
0.6
0.8
1.0
r11 , r22 for vapor
0 20 40 60 80
0.2
0.0
0.2
0.4
0.6
r12 , r21 for vapor
Large scatter, non-ideal gas, non-linearities . . . =⇒ not conclusive
The Toronto Evaporation Experiments [Ward, Stanga, ...]• forced evaporation by pumping out vapour
• steady state by resubstituting liquid =⇒ measurement of j
• temperatures in vapor measured within 1 to 5 mean free paths of interface.
• temperatures in liquid measured within 0.25mm of interface.• observed interface T-jump: up to 7.8 ◦C
prescribed datapv (Pa) Ll (mm) Lv (mm) Tbl (
◦C) Tbv (◦C)
593 4.970 18.590 26.060 25.710
measured data
Tl (◦C) Tv (
◦C) ∆T (◦C) j³kgm2 s
´−0.4 2.6 3.0 1.017× 10−3
complicated geometry & lack of detailed datax =⇒impossible to find exact fluxes/forces
Estimates (for ω = 0, planar case) [Bond&HS 2004]write interface condition as (eliminate qv)
psat (Tl)− p =r11r22 − r12r21
r22
p2πRTlj −
r12r22psat (Tl)
Tv − TlTl
evaluate for experimental data
psat (Tl) = p+r11r22 − r12r21
r220.89Pa− r12
r226.5Pa ' p− 2Pa
=⇒ liquid interface temperature determined by vapour pressure
Tl ' Tsat (pv)
=⇒ liquid heat flow ql = κl∆TLldue to ∆T between interface / boundary
=⇒ heat flow ql determines evaporation rate j ' ql/hfg
=⇒ temperature jump follows from interface heat flux Q (Tl, Tv) = ql + jhl
=⇒ condensation coefficient cannot be deduced from experiment!
=⇒ measurements not sufficiently detailed to determine rαβ!
=⇒ full alignment between theory and experiment requires more detailed measurementsand 3D simulation
Phillips-Onsager cell [Phillips et al., since 2002]
vapor
liquid
TH
TV
TvTl
TL
xV
xL
vapor
liquid
TH
T
TvTl
TL
xV
xL
TV
liquid
dry upper plate wet upper plate
pp
control: TL , TH measure: p (TH)
compute: Phillips’ heat of transfer
Q∗ = − TLpsat (TL)
dp (TH)
dTH
observation of cold to warm distillation(heat goes from warm to cold but mass goes from cold to warm)
Dry upper plate (linearized) [HS&SK&DB 2011]no convection: j = 0, conductive heat flux: Q = qv = ql = const
Q = −psat (TL)R√2πRTL
Qd (TH − TL)
cell conduction coefficient (dim.less)1
Qd=
κVκL
xLλ0+xVλ0+ r22 +
2− χ
4χβ
microscopic reference length
λ0 =κV√2πRTL
psat (TL)R
Phillips’ heat of transfer Q∗dry = −TL
psat(TL)dp(TH)dTH
Q∗dry = −
hLfgRTL
κVκL
xLλ0+ r12
κVκL
xLλ0+xVλ0+ r22 +
2− χ
4χβ
only microscopic cells xVλ0 ≤ 1 affected by resistivities rαβ, acc. coeff. χ
Wet upper plate (linearized) [HS&SK&DB 2011]convective and conductive transport
j =A
2 [C +D] +EB
∙− psat (TL)
TL√2πRTL
(TH − TL)¸
Q =B
2 [C +D] +EB
∙−psat (TL)R√
2πRTL(TH − TL)
¸Phillips’ heat of transfer
Q∗wet =hLfgRTL
1
1 +B + xL+∆
∆
£C+DE
¤xL∆B +
xL+∆∆
£C+DE
¤where
A = ZhLfgRTL
µ1
2
xVλ0+ r22
¶− r12 ,
B = ZhLfgRTL
hLfgRTL
µ1
2
xVλ0+ r22
¶−³Z + 1
´ hLfgRTL
r12 + r11
C = r111
2
xVλ0≥ 0 , D = r11r22 − r212 ≥ 0 , E =
κVκL
xL +∆
λ0≥ 0
d ln psatd lnT
= ZhLfgRTL
only microscopic cells xVλ0 ≤ 1 affected by resistivities rαβ
Heat of transfer [HS&SK&DB 2011]Q∗ is system property: Q∗dry, Q
∗wet depend strongly on thickness of bulk layers
0.0 0.2 0.4 0.6 0.8 1.0-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
relative liquid thickness =xL
xL xV+
dry upper plate
X = 0.002mm
X = 0.02mm
X = 0.07mm
X = 0.2mm
X = 7mm
Q*
relative liquid thickness =xL+D
xL+D+xV
Q*
wet upper plate
X = 0.007mm
0.001 0.005 0.010 0.050 0.100 0.500 1.000-20
-18
-16
-14
-12
-10
X = 0.07mm
X = 0.7mm
X = 7mm
X = 70mm
X — cell thickness
narrow cells (small X): dominated by interfacial processes, small Q∗dry, Q∗wet
wide cells (large X): dominated by bulk processes, large Q∗dry, Q∗wet
present measurements not sufficiently exact to determine resistivities rαβ !
Pressure and heat of transfer [HS&SK&DB 2011]theory: experiment:
0 1 2 3 4
5.3
5.4
5.5
5.6
(TH TL)/K-
p/To
rr
A
O
B
Ca
b’
upper plate drying
dry upper plate
b
wet u
pper
pla
te
Q∗dry ' 0.42 Q∗dry ' 0.9Q∗wet = 18.4 Q∗wet = 10
kink at TH = TL kink at TH = TL + 0.5K
disagreement due to:• uncertainties in T -measurement ??• different psat at upper plate (conditioning, wetting surface, . . . ) ??• ??????
non-obviuos transport modesheat goes from warm to cold
Q = jhfg + qv < 0
inverted T -profile cold to warm distillation
predicted by non-eq. TD measured by Phillips et al.??
Inverted temperature profile [Pao 1971]
vapor conductive heat flow opposite total energy flow:
J < 0 , Q < 0 , qv = Q− JhLfg > 0
equivalent to B −A hLfgRTL
> 0 or
ZhLfgRTL
>r11r12
water: 7 < ZhLfgRTL
< 20 between critical and triple points
reported values r11r12 ' 8− 10
=⇒ inverted temperature profile expected in Phillips-Onsager cell
0.000 0.001 0.002 0.003 0.004
286.0
286.5
287.0
287.5
288.0
x/mm
T/K
wet up
per p
late
dry upper p
late
Cold to warm distillation [HS&SK&DB 2011]
convective vapor flow opposite total energy flow:
J > 0 , Q < 0 , qv = Q− JhLfg < 0
equivalent to: A < 0 or
0 < xV < 2λ0
⎡⎢⎣ r12
ZhLfgRTL
− r22
⎤⎥⎦necessary criterion: xV > 0, i.e.
r12r22> Z
hLfgRTL
water: 7 < ZhLfgRTL
< 20 between critical and triple points
all reported values r12r22< 1 note: r11r22 − r12r12 > 0 =⇒ r12
r22<q
r11r22
=⇒ cold to warm distillation at best close to critical point
=⇒ experiments are close to triple point: cold to warm dist impossible!!
=⇒ possible: erroneous T -measurement or modified psat (T ) at upper plate??
Conclusions
• interface resistivities rαβ relevant mainly for microscopic flows
• experimental determination of resistivities rαβ requires:— carefully instrumented microscopic devices
— complete numerical simulation of device
• kinetic theory resistivities valid far from critical point
• molecular dynamics gives insight into resistivities [SK&DB]
• theories for resistivities towards critical point not available
• refined description of bulk phases might be necessaryx =⇒ kinetic theory, extended hydrodynamics etc
• Phillips-Onsager cell measures (macroscopic) system property Q∗
x =⇒ only mildly affected by resistivities rαβ
• cold to warm distillation appears to be impossible
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