On the (im-) possibility of cold to warm distillation Henning - - PowerPoint PPT Presentation

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On the (im-) possibility of cold to warm distillation

Henning Struchtrup

University of Victoria, Canada Signe Kjelstrup & Dick Bedeaux

NTNU Trondheim

Non-eq. condensation/evaporation [e.g., Kjelstrup & Bedeaux 2010]

mass flux j, Fourier heat flux q = −κ∂T

∂x

Interface conditions (linearized): dimensionless resistivities ˆ rαβ     

psat(Tl)−p

√

2πRTl

− psat(Tl) √

2πRTl Tv−Tl Tl

     =    ˆ r11 ˆ r12 ˆ r21 ˆ r22       j

qv RTl

   Onsager symmetry: ˆ r21 = ˆ r12 positive entropy generation: ˆ r11 ≥ 0 , ˆ r22 ≥ 0 , ˆ r11ˆ r22 − ˆ r12ˆ r21 ≥ 0 Questions: a) values of ˆ rαβ? x b) when must non-eq. interface be considered?

Interface resitivities

Kinetic theory prediction condensation coefficient ψ ≤ 1 ˆ

• rkin. theory =
• 1

ψ − 0.40044

0.126 0.126 0.294

• Compare to Hertz—Knudsen—Schrage equation

j = 2KC/E 2 − KC/E psat (Tl) √2πRTl − pv √2πRTv

• [M&S, 2001]

KC/E – condensation/evaporation coefficients ˆ r11 ≃

2−KC/E 2KC/E : KC/E ∈

• 10−3, 1
• =

⇒ ˆ r11 ∈ 1

2, 103

Phillips-Onsager cell [Phillips et al., since 2002]

control: TL , TH measure: p (TH) compute: Phillips’ heat of transfer Q∗ = − TL psat (TL) dp (TH) dTH T - difference is the sole driving force!!

non-obvious transport modes (wet upper plate)

total heat flux in vapor: ˙ Q = jhfg + qv inverted T-profile cold to warm distillation

heat ˙ Q and mass j go from warm to cold heat ˙ Q goes from warm to cold but Fourier flux qv points from cold to warm but mass j goes from cold to warm

predicted by non-eq. TD measured by Phillips et al.?? T - difference is the sole driving force!!

1-D model of Phillips-Onsager cell

Interface conditions (linearized): dimensionless resistivities ˆ rαβ     

psat(Tl)−p

√

2πRTl

− psat(Tl) √

2πRTl Tv−Tl Tl

     =    ˆ r11 ˆ r12 ˆ r21 ˆ r22       j

qv RTl

   Onsager symmetry: ˆ r21 = ˆ r12 positive entropy generation: ˆ r11 ≥ 0 , ˆ r22 ≥ 0 , ˆ r11ˆ r22 − ˆ r12ˆ r21 ≥ 0 Mass and energy balances (1-D): α = l, v (liquid, vapor) dj dx = 0 , d ˙ Q dx = d dx [jhα + qα] = 0

mass flux: j total energy flux: ˙ Q Fourier heat flux: qα = −κα∂T

∂x

enthalpy: hα

Phillips-Onsager cell [Phillips et al., since 2002]

control: TL , TH measure: p (TH) compute: Phillips’ heat of transfer Q∗ = − TL psat (TL) dp (TH) dTH

• bservation of cold to warm distillation

Dry upper plate (linearized) [HS&SK&DB 2012]

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 = κV κL xL λ0 + xV λ0 + ˆ r22 + 2 − χ 4χ microscopic reference length λ0 = κV √2πRTL psat (TL) R 0.05 mm Phillips’ heat of transfer

Q∗

dry = − TL psat(TL) dp(TH) dTH

Q∗

dry = −

hL

fg

RTL κV κL xL λ0 + ˆ r12 κV κL xL λ0 + xV λ0 + ˆ r22 + 2 − χ 4χ

• nly small cells xV

λ0

• ˆ

r12, ˆ r22, 2−χ

4χ

• affected by resist. ˆ

rαβ, acc. coeff. χ

Phillips-Onsager cell [Phillips et al., since 2002]

control: TL , TH measure: p (TH) compute: Phillips’ heat of transfer Q∗ = − TL psat (TL) dp (TH) dTH

• bservation of cold to warm distillation

Wet upper plate (linearized) [HS&SK&DB 2012]

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 = − TL psat(TL) dp(TH) dTH

Q∗

wet = hL fg

RTL 1 1 + B + xL+∆

∆

C+D

E

• xL

∆ B + xL+∆ ∆

C+D

E

• where

A = ˆ Z hL

fg

RTL 1 2 xV λ0 + ˆ r22

• − ˆ

r12 , B = ˆ Z hL

fg

RTL hL

fg

RTL 1 2 xV λ0 + ˆ r22

• −
• ˆ

Z + 1 hL

fg

RTL ˆ r12 + ˆ r11 C = ˆ r11 1 2 xV λ0 ≥ 0 , D = ˆ r11ˆ r22 − ˆ r2

12 ≥ 0 , E = κV

κL xL + ∆ λ0 ≥ 0 d ln psat d ln T = ˆ Z hL

fg

RTL

• nly small cells xV

λ0 {ˆ

r12, ˆ r22} affected by resistivities ˆ rαβ

Heat of transfer [HS&SK&DB 2012]

Q∗ = −

TL psat(TL) dp(TH) dTH

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 δ=

xL xL xV +

dry upper plate

X = 0.002mm X = 0.02mm X = 0.07mm X = 0.2mm X = 7mm

Q*

relative liquid thickness δ=

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 experiment: X ≃ 7 mm, δ ≃ 0.5

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 ˆ rαβ !

Pressure and heat of transfer [HS&SK&DB 2012]

model (kinetic theory coefficients): experiment:

1 2 3 4 5.3 5.4 5.5 5.6 (TH TL)/K

• p/Torr

A O B C

a b’

u p p e r pl a te d r y i ng

d r y u p p e r p l a t e b

wet upper plate

Q∗

dry ≃ 0.42

Q∗

wet = 18.4

Q∗

dry ≃ 0.9

Q∗

wet = 10

kink at TH = TL kink at TH = TL + 0.5 K qualitative agreement . . . BUT quantitative disagreement due to:

• uncertainties in T-measurement ??
• different psat at upper plate (conditioning, wetting surface, . . . ) ??
• values of ˆ

rαβ ??

Wet upper plate: Inverted temperature profile [Pao 1971]

vapor conductive heat flow opposite total energy flow: j < 0 , ˙ Q < 0 , qv = ˙ Q − jhL

fg > 0

equivalent to ˆ Z hL

fg

RTL > ˆ r11 ˆ r12 water: 7 < ˆ Z

hL

fg

RTL = d ln psat d ln T < 20 between critical and triple points

reported values ˆ

r11 ˆ r12 ≃ 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 upper plate d r y u p p e r p l a t e

. . . but look at the scale . . .

Wet upper plate: Cold to warm mass transfer [HS&SK&DB 2012]

convective vapor mass flow opposite total energy flow: j > 0 , ˙ Q < 0 , qv = ˙ Q − jhL

fg < 0

equivalent to: 0 < xV < 2λ0ˆ r22 ˆ Z

hL

fg

RTL

• ˆ

r12 ˆ r22 − ˆ Z hL

fg

RTL

• kinetic theory predicts:

ˆ r12 ˆ r22 = 0.43 triple point: ˆ Z hL

fg

RTL ≃ 20 = ⇒ xV < 0 cold to warm distillation impossible with kinetic theory data!!

Wet upper plate: Cold to warm mass transfer [HS&SK&DB 2012]

if observation true, what does it mean for coefficients rαβ ? rewrite previous criterion, entropy condition ˆ r11ˆ r22 − ˆ r12ˆ r12 ≥ 0 : ˆ r12 > ˆ Z hL

fg

RTL xV 2λ0 + ˆ r22

• ,

ˆ r11 ≥ ˆ r2

12

ˆ r22 combine for necessary criterion for evaporation resitivitiy ˆ r11 ≥

• ˆ

Z hL

fg

RTL 2 1 4ˆ r22 xV λ0 2 + xV λ0 + ˆ r22

• rhs has minimum at ˆ

r22|min = 1

2 xV λ0

minimum required evaporation resitivitiy ˆ r11 > 2

• ˆ

Z hL

fg

RTL 2 xV λ0 = xV 5.7 × 10−8 m ≃ 6.1 × 104 recall: ˆ r11 ≃

2−KC/E 2KC/E ∈

1

2, 103

= ⇒ impossible for Phillips’ data xV = 3.5 mm!!

Conclusions

• interface resistivities ˆ

rαβ relevant mainly for microscopic flows

• experimental determination of resistivities ˆ

rαβ requires: — carefully instrumented microscopic devices — complete numerical simulation of device

• refined description of bulk phases might be necessary

x = ⇒ kinetic theory, extended hydrodynamics etc

• molecular dynamics gives insight into resistivities [SK&DB]
• Phillips-Onsager cell measures (macroscopic) system property Q∗

x = ⇒ only mildly affected by resistivities ˆ rαβ

• cold to warm distillation appears to be impossible!!

x = ⇒ requires extreme values of ˆ rαβ

Effect of upper plate saturation pressure [HS&SK&DB 2012]

saturation pressure at the upper plate pup

sat (T∆) = pup sat (TL)

• 1 +

hL,up

fg

RTL T∆ − TL TL

• = Puppsat (TL)
• 1 + Hup

hL

fg

RTL T∆ − TL TL

• .

(1) where Pup and Hup are the ratios of saturation pressure and enthalpy between the wetted upper plate and pure water, at TL.