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Centrum for Behavior of Multiphase Systems under Super-Ambient Conditions

Electro-Diffusion flow diagnostics in liquids displaying Apparent Wall Slip effect

O. Wein, M. Večeř, V.V. Tovčigrečko, V. Sobolík :

Institute of Chemical Process Fundamentals, ASCR, Prague

• Note: The problem has been revisited by our group after almost 20 - year pause,

with improved scientific instrumentation (ED+AWS) and more realistic MT theory (3D diffusion)

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ED wall friction probes: Steady-state Inverse Operator AutoCalib ED experiment: 3D theory to Voltage-step transient Apparent Wall Slip (AWS): Viscometry & Near flow MT ED Experiment: AutoCalib MT in a Viscometric cell Combi ED+AWS results: Velocity profiles in Depleted layer

Depleted layer thickness

The purpose of this project was to study near wall velocity profiles in microdisperse liquids (polymer solutions, clay suspensions, biofluids)

by combining AWS viscometry and ED flow diagnostics

Institute of Chemical Process Fundamentals, ASCR, Prague

Centrum for Behavior of Multiphase Systems under Super-Ambient Conditions

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ED wall friction probe 1: Set-up and Primary dataCalibration data about A, cB, D are critical for the accuracy

• Note: 30% uncertainty about diffusivity corresponds to 100% uncertainty about shear rate/stress

Limiting diffusion current I is converted to MT coefficient k or the corresponding diffusion-layer thickness :

Flow: velocity profile u(z)

Pt working cathode: area A

equiv. transport length L

ED interface:voltage E(t),

current I(t) PC+

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Electrolyte solution: a cathodický depolarizer (ferro / ferri cyanide, iodide / triiodide, oxygen)

of concentration cB, diffusivity D,

and charge transfer capacity nF

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ED wall friction probe 2: Theory in DL approximationNeither shear stress nor shear rate but velocity profile u(z)

Transport model for c = c(z,x)

Solution (Leveque):

u(z)B zp uW + z (linear approximation in

viscometry, see below)

z

x L

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ED wall friction probe 3: Localized inverse operatorEffect of velocity profile is localized via diffusion thickness

Local inverse operator: 0.4

p = 0 (ideal slip), p = 0.5, p = 1 (const. shear rate)

L

• At given MT coefficient k = D/,

the local velocity u(z) at z 0.4 is independent of the shape (index p) of velocity profile:

DLA theory:

z

u

Remind:

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AutoCalib ED 1: Voltage-step transient.Cottrell (short-time) asymptote provides calibration data

• Transport model in DL approx.:

• Short-time asymptote to the complete transient:

• or: lim t0 k(t) t1/2 = (D/)1/2

• This is the well-known Cottrell (Higbie) result for transient planar diffusion through an immobile medium

Autocalibration

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10

100

1000

0.1 1 10 100 1000t, ms

I, m

kA

Complete transient in DL approx.

Steady (Leveque) DLA asymptote

Initial (Cottrell) DLA asymptote

AutoCalib ED 2: Short-time asymptote experimentally ? There are several difficulties in running short-time transient

• Solution: to treat complete I – t transient for larger times (here, t > 10 ms) and extrapolate to zero time

Faradayic (kinetic) resistance

Ohmic resistance

Amplifier threshold

Surface roughness

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AutoCalib ED 3: Full V-S transient in DL approximationWein 1981: simplifying concept of moving convective wave

• Transport model in DL approx.:

• Solution:

• Note 1: Linear transient course with t1/2k(t) vs. t1+p/2

• Note 2: The well-known steady Leveque asymptote is kL()

• Note 3: Effects of 3D diffusion neglected in DL approx.

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10

100

1000

0.1 1 10 100 1000t, ms

I, m

kA

Steady (Leveque) DLA asymptote

Initial (Cottrell) DLA asymptote

AutoCalib ED 4: Edge effects (3D diffusion)The edge effects depend on Peclet number, H L/

Complete transient in DL approximation

Complete transition, full 3D case (edge effects)

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AutoCalib ED 5: Edge effects for a still mediumThis is a 3D analogue to the Cottrell asymptote

• Transport model:

• Short-time asymptote for a still medium (Oldham 1981):P [m] - perimeter, A [m2] - area

Ideal circular probe of radius R [m] (Aoki & OesterYoung 1984):

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AutoCalib ED 6: Edge effects for a steady processThe correction N() = k()/kL() depends on H L/

• Transport model:

• Steady asymptotes for H > 1(strip, p = 1: Newman 1973, Phillips 1991, …),(disk, p = 1: Geshev 1996) (strip, p < 1: Wein 1997),(disk, p < 1: Wein 2004)

• Remind: H = L/,L = 2.64 R for disk, p=1

Disk probe, 0 < p < 1

0.0

0.2

0.4

0.6

0.8

1 10 100H

b p = 0p = 1/3p = 2/3p = 1

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AutoCalib ED 7: Edge effects for the complete transient The correction depends on H L/

• Full transport model:• Transient solution for disk, p = 1, by Geshev 1999 is wrong

• An approximation for disk, p 1 by Wein (2003): Matching of short time and steady asymptotes using the concept of convective wave (Wein 1981)

• Note: Normalization still through DLA (Cottrell, Leveque) parameters

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AWS 1: Viscometic flow of microdisperse liquids Two material functions: Slip velocity uW , Bulk shear rate

• Prototype of a viscometer (simple shear flow) with 3 primary quantities: shear stress [Pa], apparent shear rate G [s-1], hydraulic radius h [m]

U = G h = u(h) 2uW + h

h = const

zu(z) uw + z

• Viscometric determination of AWS parameters:

Several measurements of G for varied h at = const and differentiating the primary data G = G(h, )

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AWS 2: Question of actual velocity profile close to the wall

• Phenomenological explanation of the AWS effect on microscopic level: A thin layer close to wall, partially depleted of the disperse phase and, hence, displaying much lower viscosity much higher local shear rates

U = G(h,) h = u(h) 2uW + h

h

• Our task: to determine the genuine (microscale) velocity profiles within the depleted layer, using ED measurements

depleted layer

depleted layer

bulk flow

?

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Exp 1: Design of the ED viscometric cellfor simultaneous ED + AWS measurements

• 4 pairs of circular ED probes, diameters 0.2, 0.5, 1.0, 2.0 mm, mounted flush to inner surface of the viscometer wall

• Note: The probes are arranged to pairs, in order to prevent signal fluctuations due to a run-out of the rotating cylinder.

an ED flange

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Exp 2: Standard samples for calibration measurements Properties of 0÷60% aqueous glycerol solutions

• All the solutions contain equimolar amounts 25 mo/m3 of ferri-/ferro- cyanides and 3% K2SO4

y = 0.6263e-0.0082x

0.3

0.4

0.5

0.6

0.7

0 50

Glycerol concentration, wt%

D [

10-1

5m

4 s-2]

0.388.730.044Gly 60%

0.443.980.11Gly 45%

0.492.210.22Gly 30%

0.541.390.39Gly 15%

0.640.950.67Gly 0%

[10-15m4s-2] [10-6m2s-1][10-9m2s-1] 

D D  

Properties of the ED-AWS standards at 22 oC 

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Exp 3: Samples for the AWS + ED measurements Polysaccharide aqueous solns. of known AWS properties

• AWS viscometric material functions are represented by 4-param. empirical formulas:

• Solutions contain equimolar amounts 25/12.5 mol/m3 of ferri-/ferro- cyanides and 3% of K2SO4

• Their properties (D and the AWS material functions) depend only slightly on electrolyte content

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Polymer (22oC) ED, mol/m3 D [10-9m2s-1]Welan 0.25% 12.5 0.69

Welan 0.25% 25 0.69

Welan 0.50% 12.5 0.63

Welan 0.50% 25 0.63

Hercules 1% 12.5 0.68

Hercules 1% 25 0.66

Hercules 2% 12.5 0.96

Hercules 2% 25 1.01

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Exp 4: Example of the AWS viscometric data. Least square fit of primary viscometric data = (h,) using VSWork

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Exp 5: Least square fit of ED transient dataMain screen of EDWork with specifications of the records

Below: Input-output board for the single channel S_2.0:

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Exp 6: Fitted ED transients at low/high Peclet numbersR: 0.1 mm, : 4s-1 H = 4.2 | R: 1 mm, : 64s-1 H = 64

blue: Cottrell & Leveque asympt., green: DL approx., red: full 3D

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ED+AWS results1: Velocity profiles in the viscometric cellNewtonian aq.solns. as calibration standards

• Solid lines: known shear rates • Points: v(z) data for individual disk probes (0.2-2.0 mm dia)

• Note: This result confirms the edge effect correction within 1%

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ED +AWS results 2: Xanthane-type (Welan) polymer solutionsData: AWS viscometry (no ED or 25 or 12.5 mol/m3 depolarizers), ED transients with probes 0.2-2.0 mm

• Straight lines: from viscometric data: v(z) = uW[] +[] z• Individual points: via Local Inverse Operator: z = 0.4 v = 0.8DL/2

• Intersections of

macroscale AWS data (straight lines)

with microscale ones (points)

suggest

thickness of the depleted layer

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ED +AWS results 3: CMC-type (Hercules) polymer solutionsData: AWS viscometry (no ED or 25 or 12.5 mol/m3 depolarizers), ED transients with probes 0.2-2.0 mm

• Straight lines: from viscometric data: v(z) = uW[] +[] z• Individual points: via Local Inverse Operator: z = 0.4 v = 0.8DL/2

• Intersections of

macroscale AWS data (straight lines)

with microscale ones (points)

suggest

thickness of the depleted layer

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ED+AWS results 4: Estimates of depleted layer thickness

• Data for:Rotational viscometer with coaxial cylinders,

Rinner = 20mm,

= Rinner / Router = 0.94

gap h = 0.8 mm

1

10

100

1000

0.1 1 10 100 1000

Apparent shear rate N [s-1]

Dep

lete

d la

yer

thic

knes

s

[ mk

m] Wel 0.25%Wel 0.5%CMC 1%CMC 2%

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Conclusions

AutoCalibration makes it possible to safely apply the ED wall friction probes without special calibration devices

Edge effects cannot be neglected for interpreting steady-state data at low shear rates and small electrodes (H L/ < 200)

In the early (non-convective) stage of the ED transient, edge effects are larger than 1% iff t / D > (1% D/R/)-2

Using Autocalibration and Local Inverse Operator, genuine velocity profiles were determined within the diffusion layer, in particular within the depleted layer of microdisperse liquids

Thickness of the depleted layer was estimated experimentally by comparing genuine near wall velocity profiles with the viscometric AWS estimates.

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• Thanks for your attention

The project was supported by the Grant Agency of the Czech Republic

under the contracts 104/01/0545 and 104/04/0826.

Institute of Chemical Process Fundamentals, ASCR, Prague

Centrum for Behavior of Multiphase Systems under Super-Ambient Conditions