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Heat-and-Mass Transfer Relationship to Determine Shear Stress in Tubular Membrane

Systems

Nicolas Ratkovich, Pierre Bérubé and Ingmar Nopens

ASME-ATI-UIT 2010

May 18th 2010, Sorrento – Italy

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Outline

Introduction and Objectives

• Waste water treatment processes

• Reduction of fouling (two-phase flow)

• Dimensionless analysis (analogies)

Methodology

• Mass transfer (single- and two-phase flow)

• Heat-and-Mass transfer analogy

• Experimental set-up

Results and discussion

• Development of empirical model

Conclusions and future work

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Objectives

To quantify the mass transfer coefficient for two phase flow

To validate the heat-and-mass transfer analogy with the results obtained from electrochemical shear probe measurements.

To propose an empirical correlation based on heat transfer to determine the wall shear stress

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IntroductionWaste water treatment processes

• Biological removal of organic substances and nutrients (bioreactor)

• Clean water-sludge separation:

- Conventional Activated Sludge (CAS) - Gravity

- Membrane Bioreactor (MBR) - Filtration

Immersed Side-stream

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Introduction

Membrane fouling • Cake layer / pore blocking• Decreases permeate flux

Reduction of fouling• Introduction of air

- Two-phase (slug) flow

• Avoids reduction of permeate flux- Surface shear stress → scouring effect- Increases mass transfer (cake layer →

bulk region)

Slug flow• Large shear stress values • Dynamic shear stress (liquid flows down-

& up-flow)

*Taha & Cui, 2006

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Introduction

Similarities (internal flow)

Cake layer

Membrane

Flowmass

transfer

heat

transfershear stress

Dimensionless numbers

• Two physical phenomena are similar if they have the same dimensionless forms of governing differential equations and boundary conditions.

Mass transfer Wall friction Heat transfer

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Introduction

Similarities (internal and single-phase flow)

Wall friction

( )Re,dfunctionf = ( )PrRe,,dfunctionNu =

Mass transfer Heat transfer

( )ScdfunctionSh Re,,=

14.03

1

PrRe86.1

=

W

B

L

dNu

µ

µ3

1

Re62.1

=

L

dScSh

Laminar

(Re<2000) Re

64=f

Turbulent

(Re>4000)

2

9.010Re

74.5

7.3log

25.0

+

=

d

f

ε

14.0

3

1

8.0 PrRe027.0

=

W

BNuµ

µ3

1

8.0Re04.0 ScSh =

ck

dhNu =

c

p

k

c µ=Pr

f

m

D

dkSh =

fDSc

ρ

µ= 22

82

uuL

pdf w

ρ

τ

ρ=

∆=

µ

ρ du=ReDimensionless

numbers

Analogy 3

13

1

PrLe

Sc

Nu

Sh=

=

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Single-phase flowMass transfer:• Concentration polarization:

- Separation: Sludge ⇔ Solute

- Increase solute concentration near membrane surface

- Convection = Diffusion + Permeate

- Flux:

- Mass-transfer coefficient (km):

- Sh laminar correlation:

perCJdx

dCDCJ +−=

=

b

mm

C

CkJ ln

δ

Dkm =

3

1

Re62.1

==

L

dSc

D

dkSh m

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Two-phase flow

Mass transfer

• Ghosh and Cui (1999) & Zheng and Che(2006)

• Developed mass transfer correlations for:

- Falling film zone

- Wake zone

- Liquid slug zone

*Ghosh and Cui, 1999

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Heat-and-Mass transfer analogy

Developed shear stress correlation for:- Gas slug zone (falling film + wake)

- Liquid slug zone

• Analogy: Transport of momentum, mass, heat and energy

- Lewis number:

- Mass transfer coefficient:

- Heat transfer coefficient:

3

13

1

PrLe

Sc

Nu

Sh=

=

3

2

,

−

= Lec

hk

TPpTP

TPm

ρ

( )

−

−+=

25.0*

25.025.04.01.0

Pr

Pr1

155.01 I

F

F

x

xhFh

G

L

L

G

p

p

LpTPµ

µ

Gas

slug (hTP)

Liquid

slug (hL)

*Ghajar, 2010

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Experimental set-up at UBCTube diameter:

• 9.9 mm

Fluids used:

• Water + electrolyte

2 Shear probes (flow direction)

Conversion (Voltage → Shear)3

3

2

o3

5

ee

Lw

DCdπFν

V4.64µτ

=

GR

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Shear probes & Shear Stress HistogramsConversion V → τ

Correlation τ → km

Correlation Sh → τ

Correlation Nu → τ

GR

VI L =

o

2

ee

Lm

CdπFν

I4k =

2

e

3

m

D

d

0.862

kS

= Sµτ =

3

12

862.0

=

e

wem

d

D

d

dk

µ

τLiquid slug

Gasslug

3

2

561.1Sh

d

D

e

w

µτ =

3

2 Pr

561.1Nu

Sc

d

D

e

w

=

µτ

3

1

Pr

=

ScNuSh

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Electrochemical measurements

Single-phase flow

Shear stress Sherwood number

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 200 400 600 800 1000 1200

Re

Sh

ear

str

es

s (

Pa

)

Theory

Shear probes

0

5

10

15

20

25

30

35

0 200 400 600 800 1000 1200 1400

Re

Sh

Experimental data

Lévêque correlation

This work

3

1

Re62.1

=

L

dScSh

3

1

Re1.495

=

L

dScSh

Re

82

uw

ρτ =

Difference of 8 % between theory and experimental data

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Empirical model

Two-phase flow

• Gas slug

• Liquid slug

• Correction factors:

- Coalescence

- Bubble length

- Hydraulic diameter

- Transition regime (calibration under laminar conditions)

- &

33

2,Pr

561.1Lls

L

L

e

lsw NuSc

d

Dφ

µτ

=

33

2,Pr

561.1TPgs

TP

TP

e

gsw NuSc

d

Dφ

µτ

=

gsφ lsφ

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y = 1508.7567x-1.1957

R2 = 0.9113

y = 545.7382x-0.2945

R2 = 0.9461

0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200

ReL

a1,l

s R

eL

a2,ls

0

0.4

0.8

1.2

1.6

2

2.4

2.8

a1,g

s R

eL

a2,g

s

Liquid slug

Gas slug

Power (Gas slug)

Power (Liquid slug)

Empirical model

Correction factor

• Power law expression

• Based on experimental measurements

Final expression

• Liquid slug

• Gas slug

( ) 33295.0

, Re900.48 LLlsw Nu−

=τ

( ) 33196.1

, Re741.138 TPLgsw Nu−

=τ

2Re1

a

La=φ

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Empirical modelReSG

Rem

0

20

40

60

80

100

120

140

0 5 10 15 20 25 30 35 40 45

ReSG

a1

,ls R

eS

Ga

2,l

s

0

0.4

0.8

1.2

1.6

2

2.4

2.8

a1

,gs R

eS

Ga

2,g

s

Liquid slug

Gas slug

Power (Gas slug)

Power (Liquid slug)

0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200 1400 1600 1800

Rem

a1,l

s R

em

a2,l

s

0

0.4

0.8

1.2

1.6

2

2.4

2.8

a1

,gs R

em

a2,g

s

Liquid slug

Gas slug

Power (Gas slug)

Power (Liquid slug)

Reff

Resf

0

20

40

60

80

100

120

140

186 188 190 192 194 196 198 200

Reff

a1

,ls R

eff

a2,l

s

0

0.4

0.8

1.2

1.6

2

2.4

2.8

a1,g

s R

eff

a2,g

s

Liquid slug

Gas slug

Power (Gas slug)

Power (Liquid slug)

0

20

40

60

80

100

120

140

0 500 1000 1500 2000 2500 3000

Resf

a1

,ls R

esfa

2,ls

0

0.4

0.8

1.2

1.6

2

2.4

2.8

a1

,gs R

esfa

2,g

s

Liquid slug

Gas slug

Power (Gas slug)

Power (Liquid slug)

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Conclusions

Shear stress values were obtained from shear probes (electrochemical method) using the Sherwood number

Using the analogy between heat-and-mass transfer an empirical correlation was developed for two-phase flow to determine the wall shear stress:• Two zones: liquid (L) and gas slug (TP)• Predictions:

- Single phase flow is acceptable (10 % error)- Two-phase flow: error up to 60 %

> Heat transfer coefficient correlation for TP has errors up to 30%> The correlation is mainly designed for turbulent regime> Common membrane operation is in laminar-transition regime

Analogies are determined mainly for turbulent regime; operation of tubular air lift membranes is in laminar-transition regime

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Future work

Non-Newtonian liquids (i.e. sludge)• Use of CMC as a non-Newtonian liquid to mimic the

properties of Sludge.• Viscosity (flow in a pipe)

• Reynolds and Prandtl number

• Nusselt number correction

• Viscosity correction:

B

SLLMR

du

µ

ρ=Re

3

1

4

13

+=−

n

n

Nu

Nu

New

Newnon

14.0

W

B

µ

µ

11

8

4

13−−

+=

n

SL

n

Wd

u

n

nKµ

18

4

13−

+=

n

SL

n

Bd

u

n

nKµ

c

Bp

k

c µ=Pr

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Acknowledgement

MBR-TRAIN is a Marie Curie Host Fellowship for Early

Stage Research Training supported by the European

Commission under the 6th Framework Programme

(Structuring the European Research Area - Marie Curie

Actions)

Contract No. MEST-CT-2005-021050

Duration: 01/01/06 - 31/12/09

MBR-TRAIN is part of the MBR-NETWORK Cluster

More info: www.mbr-network.eu and www.mbr-train.org

Funding for the infrastructure used to measure surface shear forces was provided by the Natural Science and Engineering Research Council of Canada (NSERC).