xxxx - Lorentz Center · Hertz—Knudsen—Schrage equation j = 2 ... Onsager symmetry: r ......

24
xxxx Evaporation and condensation in non-equilibrium Henning Struchtrup J.P. Caputa University of Victoria, Canada Signe Kjelstrup & Dick Bedeaux NTNU Trondheim

Transcript of xxxx - Lorentz Center · Hertz—Knudsen—Schrage equation j = 2 ... Onsager symmetry: r ......

Page 1: xxxx - Lorentz Center · Hertz—Knudsen—Schrage equation j = 2 ... Onsager symmetry: r ... =⇒needs microscopic modelling and/or experiment.

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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

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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

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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?

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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

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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)

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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)

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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]

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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

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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?

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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

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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

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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

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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)

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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

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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

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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 !

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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, . . . ) ??• ??????

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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.??

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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

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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??

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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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