Hydrodynamic flow conditions through permeable walls

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Hydrodynamic flow conditions through permeable walls

S. Vollmar, J. A. M. S. Duarte

To cite this version:

S. Vollmar, J. A. M. S. Duarte. Hydrodynamic flow conditions through permeable walls. Journal de

Physique I, EDP Sciences, 1992, 2 (8), pp.1565-1569. �10.1051/jp1:1992228�. �jpa-00246641�

Classification Physics Abstracts 47.55

Hydrodynamic flow conditions through permeable walls

S. Vollmar and J. A. M. S. Duarte

Institute of Theoretical Physics, Cologne University, W-5000 K61n 41, Germany

(Received

and accepted 27 Aprfl

1992)

Abstract. The flow through permeable walls is studied for _a variety of hole dimensions, spacings, and boundary conditions by the method ofhydrodynamic cellular automata. The flux

measured _on the contact section of two adjacent channels is found to scale for all studied hole sizes from I to 128, regardless of whether the flow is forced by _an imposed _pressure difference

between both channels _or by the friction of the boundary layers in each of them.

In the _course of industrial and Scientific

development during

the last 200 _years the

problem

of the

separation,

concentration and

purification

of molecular mixtures has

inspired techniques

such _as

distillation, precipitation, crystallization, extraction, adsorption,

ion

exchange,

etc.

Semipermeable

membranes _{are a} rather _recent tool for the named

separation problem

that is of

major importance

for the chemical _as well _as the food and

drug

industries. The tech- nical term "membrane" refers to a medium that is suitable to solve _a

given

_mass

separation

task without

damaging

_or

chemically altering

the

constituents,

which is _a

particularly

useful property.

According

to Strathmann

[I],

the technical and commercial

impact

of membrane

separation

_processes has become

"significant".

It is the

object

of this _{paper to}

enlarge

the basis for further membrane Studies

by

_a

hydro- dynamic Study

of

permeable

walls

adding

to the work that has

already

been done

by

Brosa and co-workers

(see

_{[2, 3]} for

details). Pioneering

studies of

semi-permeable

walls have not been

entirely

successful due to the _use of Systems of

inadequate

size [4], or

possibly

because the membrane condition

implementation

does not translate the full

complexity

of real membrane

technology.

In

fact,

membranes _{can now} separate fluid components

keeping

the normal

velocity equal

_as first done in [2],

translating

^this component

along

^{the membrane}

[I],

_or even

involving

an active transport of the fluid from _one to the other side

through complex

chemical reactions

[I].

It is

only

_now, that _a sounder grasp of the system ^{sizes and simulation} times

required

for

a

realistically

successful _use of

hydrodynamic

cellular automata is

available,

that the founda- tions _are

being

settled for studies of

semi-permeable

elements in their wide

variety

_[5]. Our

study

concentrates,

therefore,

_on

permeability properties

and _{not on} its

differentiating

effects in two-fluid systems.

The

typical

^lattice

geometry

was a lattice of

length

L

= 1920 in lattice constants with two

symmetrical

channels of

heigth

H

= 660 each. The

permeable

section of the middle wall had

1566 JOURNAL DE PHYSIQUE I N°8

a

length

of LM

=

L/2

and _was centered

starting

at

L/4.

See

figure

I for _an iuustration of the lattice geometry. ^We use the

hydrodynamical

cellular automata

algorithm ("lattice gas"

in the form of reference [5].

Fig. 1. Two symmetric channels communicating through _an intermediate permeable section offour segments. Average velocities _were taken _on 60 by 64 cells.

As described in [2], ^{Brosa and cc-workers used} a

boundary

condition that forces

particles through

the

permeable

section

by

_means of _a strong _pressure

gradient.

From _an

equal

_pressure

gradient

at the

beginning

two different _{ones are} reached at the downstream end of

respectively

the _upper and lower channel.

Adaptation

to this variable condition is achieved in normal

flow

through

the deviation of the _excess fluid from the lower to the _upper

velocity

channel.

Taking

^this

procedure

to its

logica1extreme, by putting

a wall _at the downstream end of _one channel and

initializing

_average zero

velocity there,

_we found that instantaneous vector fields show

big

vortices

acting

as a bulk and

imposing

the deviation of the flow to the other channel.

For further measurements, we chose initial

velocity

values that _are in comfortable

agreement

with the

typical

values where flow is

incompressible leading

to _a maximum

speed

which h

appropriate

^for

obtaining

measurements at the

permeable

section. Our

typical

flow

through

the

membrane,

"the

throughput"

in the

terminology

of references [2, 3], was varied between Gil _to 30il of the flow _average in the _upper upstream channel in most of _our combinations of

upper and lower maximum

speed.

For _a very

long

_run, 400000

iterations, averaging

_every 500 time steps, we find the _pressure

gradient

of

figure

2.

In contrast to the "forced"

boundary

conditions described

by Brosa,

our

alternative,

the

"free"

boundary conditions, impose

the _same Poheuille

profile

at the upstream ^{and downstream} end of each channel

(see

_{[5] for} a more detailed

description).

The channels _are initialized with

different velocities thus

particles

_are

"dragged" through

^the

permeable

section into the channel with the

higher

_average

velocity by

_means of friction between the upper and lower

boundary layer.

As could be

expected

the

throughput

in this _case h

drastically

about six times smaller

than for the

corresponding

combination of

speeds

in the forced flow _case.

There have been attempts in the past to

analyze carefully

the evolution of the

layer

between

Beginning

of Channel Middle of Channel

882 882

m

m m m

8?8 e?8 .

~ .876 _~

t$ t$

@ _~~~

© _.

CQ 872 ^.

870 .

8?o . .

m

888 ₈₆₈ . _.

m

866 866

y y

1568 JOURNAL DE PHYSIQUE I N°8

heigth

H constant does

apparently

not affect the flux. The

(b)-type permeable

section with D ₌ 8

(straight)

^and D

= 16

(dashed), respectively,

_was used.

lLlH

n2LlH *lL2H o2L2H

/ / /°

/ /

,'

/ / _/

/ / /

~ ^/

~ ^/

~ /

'

Q ~

i~~ 100000

/ / soooo

o

-5°°°°o

sooo isooo 25000

time

Fig. 3. Accumulated flux through the permeable section when the channel dimensions change. The dashed line refers to _a D

= 16 and the solid _one to _a D

= 8 geometry, respectively.

For

measuring

the flow

through

the

(a),type permeable

section

(scattering sites)

the

length

of the

permeable

section _was chosen _so that the number of fluid sites _was constant at 640 for each D. The

length

L of the lattice _was then

adjusted

to twice the

length

of the

permeable

section.

The

graph

of the flux _versus

D,

for D

- co

(figure 4), depicts

_our results for different

boundary conditions, geometries

of

permeable

sections and velocities. While _our

computer

simulations also allow for small

D, reality normally corresponds

to D _{- co.} The _square markers represent the forced

boundary

conditions with full _squares for the

(a)-type geometry (D, I, D...)

and _{open squares} for the

(b)-type (D, D, D...).

Free channel

boundary

conditions for two different fluxes _are marked

by

stars and fun circles. As _can be _seen from

figure

4, the data

collapses

for all

D's,

for all variations of the _upper and lower inital velocities and for both

boundary

conditions in the transition of the

permeable

wall to a full

opening

that allows

unhindered flow. This data

scaling

has _a

significant physical importance,

in that it shows

that,

whatever the level of unhindered flux for D _{- co}

maybe (as,

_e-g-,

through

the

openings

that

are D ₌ 128 lattice sites

wide)

the ratios of the various

spacings

down to _even D

= I follow

the _{same curve} within statistical _errors. It also shows that whatever the mechanism that h

mainly responsible

for the

flow,

the _same

unifying scaling applies.

To obtain _a

velocity

field _we divided _our lattice in cells of size 64

by

60. We summed the y-components ^{of the}

velocity

vectors in the

regime

of the first half of the

permeable

section and the second half. For the

(a)-type

membrane geometry

(D

fluid

sites,

D obstacle sites

,

D fluid

sites...)

and the

forcing boundary

conditions _we found _a

higher

total

velocity

for the second half of the

permeable

section than for the first half. This result _was

irrespective

of the value of D. The

(b)-type

membrane geometry,

however,

shows this

property

^for _very ^small D

1

D,D,D...

_.

D,I,D...

~ i-o

, ^{, o .}

~

. ~

QZ fl

o 6

°n~

$

.~

$

~

[~

fi

.*a

t>h

i

~

~

~ z

~ iZ

~

0 20 40 60 80 loo 120 140

D

Fig. 4. Normalized flux plot for various hydrodynamic flow conditions _{as a} function of D squares refer to the forced boundary conditions,while _* ^refer to the free channel _case

(with

_average peak

velocities 0A and 0.I in each

channel)

and _e to _a 0.35 to 0.20 peak velocity ratio for the free channel

case also

(in

both these 2 _{cases a} D, D, D... geometry was

used).

only;

the

opposite effect,

I-e- _a

higher

total

velocity

in the first half of the

permeable

section than in the second

half,

_was observed for

larger

values of D.

Acknowledgements.

We _are indebted for _a time grant on the NEC SX3

Ill

and to Dietrich Stauffer for

stimulating suggestions.

We _are

grateful

to G. A.

Kohring

for _many

illuminating

dhcussions and _numerous

algorithmic suggestions.

One of _us

(J.

A. M. S.

D.)

is indebted to JNCIT

(Portugal)

for _a senior research

fellowship.

References

[ii

H. Strathmann, C. A. Costa and J. S. Cabral Eds., Chromatographic and Membrane Processes in Biotechnology,

(Kluwer

Academic Publishers, The Netherlands

1991)

_pp. 153-175.

[2] Brosa U., Kfittner C., Werner U., J. Stat. Phys. 60

(1990)

875.

[3] ^Brosa U., J. Phys. France 51

(1990)

1051.

[4] Flekkoy E., Feder J., Jossang T., preprint Oslo University

(1991).

[5] Kohring G-A-, J. Stat. Phys. 63

(1991)

411; ^J. Phys. II France1

(1991)

_{87, 594;} Int. J. Mod.

Phys. C 2

(1991)

755.

[6] Schlichting H., Grenzschichtheorie

(Braun

Verlag, Karlsruhe

,

1951).

[7] Roache P-J-, Computational Fluid Dynamics

(Hermosa,

Albuquerque,

1972).

[8] ^Lock R-C-, Quarterly J. Mean. Appl. Math. 4

(1951)

42.

JOURNAL DE PHYSIQUE I -T 2. N'S, AUGUST lW2 56