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

xxxx 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 r αβ       p sat ( T l ) − p √ j r 11 ˆ r 12 ˆ   2 πRT l          =      q v − p sat ( T l ) T v − T l √ r 21 ˆ r 22 ˆ RT l T l 2 πRT l Onsager symmetry: r 21 = ˆ ˆ r 12 positive entropy generation: r 11 ≥ 0 , ˆ ˆ r 22 ≥ 0 , ˆ r 11 ˆ r 22 − ˆ r 12 ˆ r 21 ≥ 0 Questions: a) values of ˆ r αβ ? x b) when must non-eq. interface be considered?

Interface resitivities Kinetic theory prediction condensation coefficient ψ ≤ 1 � � 1 ψ − 0 . 40044 0 . 126 r kin. theory = ˆ 0 . 126 0 . 294 Compare to Hertz—Knudsen—Schrage equation � p sat ( T l ) � 2 K C/E p v √ 2 πRT l √ 2 πRT v j = − 2 − K C/E [ M����&S����� , 2001] K C/E – condensation/evaporation coefficients � � � 1 2 , 10 3 � 2 −K C/E 10 − 3 , 1 r 11 ≃ ˆ 2 K C/E : K C/E ∈ = ⇒ r 11 ∈ ˆ

Phillips-Onsager cell [Phillips et al., since 2002] control: T L , T H measure: p ( T H ) compute: Phillips’ heat of transfer T L dp ( T H ) Q ∗ = − p sat ( T L ) dT H T - difference is the sole driving force!!

non-obvious transport modes (wet upper plate) ˙ total heat flux in vapor: Q = jh fg + q v inverted T -profile cold to warm distillation heat ˙ heat ˙ Q and mass j go from warm to cold Q goes from warm to cold but Fourier flux q v 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 r αβ       p sat ( T l ) − p √ j r 11 ˆ r 12 ˆ   2 πRT l         =       q v − p sat ( T l ) T v − T l √ r 21 ˆ r 22 ˆ RT l T l 2 πRT l Onsager symmetry: r 21 = ˆ ˆ r 12 positive entropy generation: r 11 ≥ 0 , ˆ ˆ r 22 ≥ 0 , ˆ r 11 ˆ r 22 − ˆ r 12 ˆ r 21 ≥ 0 Mass and energy balances (1-D): α = l, v (liquid, vapor) dx = 0 , d ˙ dj dx = d Q dx [ jh α + q α ] = 0 mass flux: j total energy flux: ˙ Q Fourier heat flux: q α = − κ α∂T ∂x enthalpy: h α

Phillips-Onsager cell [Phillips et al., since 2002] control: T L , T H measure: p ( T H ) compute: Phillips’ heat of transfer T L dp ( T H ) Q ∗ = − p sat ( T L ) dT H observation of cold to warm distillation

Dry upper plate (linearized) [HS&SK&DB 2012] no convection: j = 0 , conductive heat flux: ˙ Q = q v = q l = const Q = − p sat ( T L ) R ˙ √ 2 πRT L Q d ( T H − T L ) cell conduction coefficient (dim.less) 1 = κ V x L + x V r 22 + 2 − χ + ˆ Q d κ L λ 0 λ 0 4 χ microscopic reference length √ 2 πRT L λ 0 = κ V p sat ( T L ) R � 0 . 05 mm T L dp ( T H ) Phillips’ heat of transfer Q ∗ dry = − p sat ( T L ) dT H h L κ V x L fg + ˆ r 12 RT L κ L λ 0 Q ∗ dry = − κ V x L + x V r 22 + 2 − χ + ˆ κ L λ 0 λ 0 4 χ � � λ 0 � only small cells x V r 22 , 2 − χ affected by resist. ˆ r αβ , acc. coeff. χ r 12 , ˆ ˆ 4 χ

Phillips-Onsager cell [Phillips et al., since 2002] control: T L , T H measure: p ( T H ) compute: Phillips’ heat of transfer T L dp ( T H ) Q ∗ = − p sat ( T L ) dT H observation of cold to warm distillation

Wet upper plate (linearized) [HS&SK&DB 2012] convective and conductive transport � � A − p sat ( T L ) √ 2 πRT L j = ( T H − T L ) 2 [ C + D ] + EB T L � � B − p sat ( T L ) R ˙ √ 2 πRT L Q = ( T H − T L ) 2 [ C + D ] + EB T L dp ( T H ) Phillips’ heat of transfer Q ∗ wet = − p sat ( T L ) dT H wet = h L 1 fg Q ∗ � C + D � 1 + B + x L +∆ RT L ∆ E � C + D � x L ∆ B + x L +∆ ∆ E where � 1 � Z h L x V fg A = ˆ + ˆ r 22 − ˆ r 12 , RT L 2 λ 0 � 1 � � � h L Z h L h L x V fg fg fg B = ˆ ˆ + ˆ r 22 − Z + 1 r 12 + ˆ ˆ r 11 RT L RT L 2 λ 0 RT L 1 x V 12 ≥ 0 , E = κ V x L + ∆ r 2 C = ˆ r 11 ≥ 0 , D = ˆ r 11 ˆ r 22 − ˆ ≥ 0 2 λ 0 κ L λ 0 Z h L d ln p sat = ˆ fg d ln T RT L λ 0 � { ˆ only small cells x V r 12 , ˆ r 22 } affected by resistivities ˆ r αβ

Heat of transfer [HS&SK&DB 2012] Q ∗ = − T L dp ( T H ) is system property p sat ( T L ) dT H Q ∗ dry , Q ∗ wet depend strongly on thickness of bulk layers wet upper plate dry upper plate 0.0 - 10 X = 0.002mm X = 0.02mm - 0.5 - 12 X = 0.007mm - 1.0 X = 0.07mm - 14 X = 0.07mm Q * Q * - 1.5 X = 0.7mm - 16 - 2.0 X = 0.2mm X = 7mm - 18 - 2.5 X = 7mm X = 70mm - 20 - 3.0 0.0 0.2 0.4 0.6 0.8 1.0 0.001 0.005 0.010 0.050 0.100 0.500 1.000 xL + D xL relative liquid thickness δ = relative liquid thickness δ = + xL + D+ xL xV xV X — cell thickness experiment: X ≃ 7 mm , δ ≃ 0 . 5 narrow cells (small X ) : dominated by interfacial processes, small Q ∗ dry , Q ∗ wet (large X ) : dominated by bulk processes, large Q ∗ dry , Q ∗ wide cells wet present measurements not sufficiently exact to determine resistivities ˆ r αβ !

Pressure and heat of transfer [HS&SK&DB 2012] model (kinetic theory coefficients) : experiment: wet upper plate C u p a p e r pl a 5.6 te d r y i ng A e a t p l e r p p u r y d p/Torr 5.5 b’ O b 5.4 B 5.3 0 1 2 3 4 - (TH TL)/K Q ∗ Q ∗ Q ∗ Q ∗ dry ≃ 0 . 42 wet = 18 . 4 dry ≃ 0 . 9 wet = 10 kink at T H = T L kink at T H = T L + 0 . 5 K qualitative agreement . . . BUT quantitative disagreement due to: • uncertainties in T -measurement ?? • different p sat at upper plate (conditioning, wetting surface, . . . ) ?? • values of ˆ r αβ ??

Wet upper plate: Inverted temperature profile [Pao 1971] vapor conductive heat flow opposite total energy flow: Q < 0 , q v = ˙ ˙ Q − jh L j < 0 , fg > 0 equivalent to h L > ˆ r 11 fg ˆ Z RT L r 12 ˆ h L water: 7 < ˆ RT L = d ln p sat fg d ln T < 20 between critical and triple points Z reported values ˆ r 11 r 12 ≃ 8 − 10 ˆ inverted temperature profile expected in Phillips-Onsager cell 288.0 287.5 wet upper plate e t a l p T/K r e 287.0 p p u y r d 286.5 286.0 0.000 0.001 0.002 0.003 0.004 x/mm . . . 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: Q < 0 , q v = ˙ ˙ Q − jh L j > 0 , fg < 0 equivalent to: � � Z h L 0 < x V < 2 λ 0 ˆ r 22 r 12 ˆ fg − ˆ h L r 22 ˆ RT L ˆ fg Z RT L kinetic theory predicts: r 12 ˆ = 0 . 43 r 22 ˆ triple point: Z h L fg ˆ ≃ 20 RT L = ⇒ x V < 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 r αβ ? rewrite previous criterion, entropy condition ˆ r 11 ˆ r 22 − ˆ r 12 ˆ r 12 ≥ 0 : � x V � Z h L r 2 r 11 ≥ ˆ fg r 12 > ˆ 12 ˆ + ˆ r 22 , ˆ RT L 2 λ 0 r 22 ˆ combine for necessary criterion for evaporation resitivitiy � � 2 � � � x V � 2 Z h L 1 + x V fg ˆ r 11 ≥ ˆ + ˆ r 22 RT L 4ˆ r 22 λ 0 λ 0 x V r 22 | min = 1 rhs has minimum at ˆ 2 λ 0 minimum required evaporation resitivitiy � � 2 x V h L x V fg ˆ 5 . 7 × 10 − 8 m ≃ 6 . 1 × 10 4 r 11 > 2 ˆ Z = RT L λ 0 � 1 2 , 10 3 � 2 −K C/E recall: ˆ r 11 ≃ 2 K C/E ∈ = ⇒ impossible for Phillips’ data x V = 3 . 5 mm !!

Conclusions • interface resistivities ˆ r αβ relevant mainly for microscopic flows • experimental determination of resistivities ˆ r αβ 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 = r αβ • cold to warm distillation appears to be impossible!! x = ⇒ requires extreme values of ˆ r αβ

Effect of upper plate saturation pressure [HS&SK&DB 2012] saturation pressure at the upper plate � � � � h L, up h L T ∆ − T L T ∆ − T L fg fg p up sat ( T ∆ ) = p up sat ( T L ) 1 + = P up p sat ( T L ) 1 + H up . (1) RT L T L RT L T L where P up and H up are the ratios of saturation pressure and enthalpy between the wetted upper plate and pure water, at T L .