flow between parallel plates - femlabrev2006

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Flow Between Parallel Plates – Modified from the COMSOL ChE Library module.

Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall !!"

The Naier!Sto"es E#uations

#he $avier%&tokes e'(ations describe flow in visco(s fl(ids thro(gh moment(m balances foreach of the com)onents of the moment(m vector in all s)atial dimensions. #hey also ass(me that

the density and viscosity of the modeled fl(id are constant, which gives rise to a contin(ity

condition.

#he *ncom)ressible $avier%&tokes a))lication mode in C+M&+ is somewhat more general

than this and is able to acco(nt for arbitrary variations in viscosity and small variations indensity- say thro(gh the o(ssines' a))ro/imation.

#his section )rovides a s(mmary of the )artici)ating e'(ations and bo(ndary conditions for oneof the most commonly (sed )hysical descri)tions (sed in mathematical modeling. Following this

is also a descri)tion of how the $avier%&tokes e'(ations can be scaled.

E$%&T'ONS

#he e'(ations in the $avier%&tokes a))lication mode are defined by E'(ation 0%1 for a variableviscosity and constant density. #he moment(m balances and contin(ity e'(ation form a

nonlinear system of e'(ations with three and fo(r co()led e'(ations in 2 and 32, res)ectively.

40%15

where η denotes the dynamic viscosity 4M %1 #%1 5, u the velocity vector 4 #%15, ρ the density of

the fl(id 4M %35, p the )ress(re 4M %1 #%5 and F is a body force term 4M % #%5 s(ch as gravity.

#he first e'(ation is the momentum balance, and the second is the equation of continuity for

incom)ressible fl(ids.

Remember from yo(r te/t that symbol, , is a mathematical o)erator. #he gradient or 6grad7 of a

scalar field s(ch as )ress(re is defined as8

z

p

y

p

x

p p

∂

∂+

∂

∂+

∂

∂=∇ " (i E'(ation 3."%11

*n addition the dot )rod(ct of the symbol, , with another vector is called the divergence or 6div7

of a vector v.

z

v

y

v

x

v z y x

∂

∂+

∂

∂+

∂

∂=⋅∇ E'(ation 3."%1

1

http://var/www/apps/conversion/tmp/scratch_5/chem31_ug_momentum2.htm#wp378760

http://var/www/apps/conversion/tmp/scratch_5/chem31_ug_momentum2.htm#wp378760



For constant density fl(ids at steady%state the contin(ity e'(ation gives

!=∂

∂+

∂

∂+

∂

∂=⋅∇

z

v

y

v

x

v z y x E'(ation 3."%0

#he $avier%&tokes E'(ation 4constant density and viscosity5 for rectang(lar coordinates from the

9eanko)lis book is given as8

( ) )

∇++−∇=∇⋅+∂

∂= µ ρ ρ ρ ρ p

t Dt

D E'(ation 3.:%3;

Boundary Conditions

#he bo(ndary conditions can be any of a n(mber described below. For an im)osed velocity, the

velocity vector normal to the bo(ndary can be s)ecified by8

!u=⋅nu 40%5

which is denoted as the 'nflow*Outflow boundary +ondition. *n the above e'(ation n is a (nitvector that has a direction )er)endic(lar to a bo(ndary or normal to a bo(ndary. For e/am)le in

hori<ontal )i)e flow the inlet velocity is )er)endic(lar to the cross sectional area of the )i)e. =ecan instead im)ose a certain )ress(re in the Outflow*Pressure boundary +ondition8

! p p = 40%35

#here is a s)ecial bo(ndary condition that combines both these descri)tions. #his is the Normal

flow*Pressure or ,strai)ht!out- boundary +ondition, which sets the velocity com)onents in

the tangential direction to <ero, and sets the )ress(re to a s)ecific val(e. *t is )rovided to sim(latechannels that are long in length where flow is ass(med to have stabili<ed so that all velocity

occ(rring in the tangential direction is negligible. *m)osing this condition removes the necessity

of sim(lating this length.

!and p p ==⋅ .tu 40%05

=here the (nit vector t, is tangent to a bo(ndary.

#he Sli/*Symmetry +ondition states that there are no velocity com)onents )er)endic(lar to a

bo(ndary8!=⋅nu 40%>5

#he Neutral boundary +ondition states that trans)ort by shear stresses is <ero across a

bo(ndary. #his bo(ndary condition is denoted ne(tral since it does not )(t any constraints on the

velocity and states that there are no interactions across the modeled bo(ndary.

40%"5

#he No!sli/ boundary +ondition eliminates all com)onents of the velocity vector8!=u 40%:5



SC&L'N0

Many modeling domains may be large in one dimension as o))osed to the others. #his can create

(nreasonably large com)(tational )roblems, which may not be able to be solved. &caling the

e'(ations and dimensions of the geometry can create a more reasonable model si<e.

#he scaling )rocess ideally involves s)ecifying the $avier%&tokes e'(ations in a dimensionless

form. *n this case, we can introd(ce a characteristic length, d , and a characteristic velocity,u!.

For flow in a circ(lar t(be, d is often taken to be the t(be diameter and u! the average velocity.

*n this case, new dimensionless variables are defined according to8

40%?5

where p ! denotes a reference )ress(re. #he moment(m e'(ations, in dimensionless form,

become

40%;5

where the differential o)erators are also e/)ressed in dimensionless form. #his im)lies that the bo(ndary conditions and domain geometry m(st also be defined in dimensionless form. #he

inverse of the dimensionless n(mbers within the s'(are brackets are the Reynolds and Fro(den(mbers, res)ectively.

#he *ncom)ressible $avier%&tokes a))lication mode also comes with s)ecial element ty)es with

different sha)e f(nctions. #hey incl(de the discontin(o(s element 4shdisc5 and the b(bble

element 4shbub5. #hey are s(mmari<ed in the Application Mode Implementation cha)ter (nder

the section for Incompressible Navier-Stokes.

#he $avier%&tokes e'(ations not only describe all ty)es of $ewtonian incom)ressible flow, b(t,

theoretically, can also describe t(rb(lent flow. However, modeling of t(rb(lent flow with the

$avier%&tokes e'(ations is im)ractical in most engineering a))lications, since it re'(ires that

even the smallest eddies are to be resolved, which creates an (nreasonably large n(mber of nodesin the discreti<ation )rocess. F(rthermore, flow cannot be said to be stationary since the eddies

move mostly randomly thro(gh the flow.

#hese ty)es of time%de)endent sim(lations are very demanding in the n(mber of com)(tational

o)erations and memory re'(ired, and they are too large to be handled by most com)(ters. *t is

therefore necessary to (se sim)lified models for the modeling of t(rb(lent flow. +ne s(ch

a))roach is incl(ded in the k%ε a))lication mode available in the mod(le.

3



@lso note that the Chemical Engineering Mod(le contains an a))lication mode for non%

$ewtonian fl(ids with )redefined viscosity models.

More information abo(t the $avier%&tokes @))lication Mode can be fo(nd in the Modeling

9(ide, Fl(id Mechanics Cha)ter.

#he following is a sim)le e/am)le of (se of the *ncom)ressible $avier%&tokes a))lication mode

in the Chemical Engineering Mod(le. *t will ac'(aint yo( with the different men(s ando)erations, as well as )rovide a st(dy of develo)ing laminar flow.

0



Steady!State Flow Between Parallel Plates

1E2am/le 3.4!5 in 0ean"o/lis6

*n this )roblem yo( will model the steady%state flow of a fl(id

thro(gh vertical )lates, with infinite width and having a s)acing

of !.! m as shown in the adAacent fig(re. #he vertical directioncoordinate is y and the hori<ontal direction coordinate is /.

1. #he initial )roblem will be to s)ecify a (niform velocity of

!.! mBs at the entrance of the )lates and determine avelocity )rofile. #his )roblem will show a (niform

velocity )rofile that transitions into f(lly develo)ed flow

which is a )arabolic velocity )rofile. Form(late amoment(m balance for this )roblem and obtain the

analytical sol(tion for the velocity )rofile.

. *n the second )roblem yo( will s)ecify a )ress(re dro) and

then calc(late a velocity )rofile. *n this case the velocity

)rofile will be f(lly develo)ed from the start of thegeometry.

Modeling using the Graphical User Interface

1. &tart C+M&+ M(lti)hysics. *n the Model Nai)ator , click the New )age

3. &elect Chemi+al En)ineerin) Module, Momentum

Balan+e, 'n+om/ressible Naier!Sto"es, &teady%stateanalsis

0. Click O7.

Options and Settings#he first ste) in the modeling )rocess

is to create a tem)orary database for the in)(t data. #he mod(le s(ggests

(nits in &* and e/)ects yo( to (se

consistent set of (nits. *n this case, all(nits are &* (nits. 2efine the following

constants in the Constants dialog bo/

in the O/tion men(. $ote this is not

water and it has a (niform inlet

velocity of ×1!% mBs, viscosity of

!.!1 kgB4m s5 and density of 1!!!

kgBm3.

N&ME E8P9ESS'ON

rho 1e3

eta 1e%

v! e%

>

/

y



0eometry Modelin) 1Create Parallel Plates6

1. Press the Shift key and click the

9e+tan)le*S#uare b(tton to bring () themen( for the RectangleB&'(are.

. #y)e the val(es shown in the adAacentfig(re for the rectangle dimensions.

3. Click the :oom E2tents b(tton in theMain toolbar.

Subdomain Settings

&elect Subdomain Settin)s from the Physi+s

men(. 2efine the )hysical )ro)erties of the

fl(id, in this case viscosity and density,

according to the table below.

Boundary Conditions

&elect Boundary Settin)s from the Physi+smen( and enter the bo(ndary conditions according to the following table8

BO%N;&9< 5=> ? 3

o(ndary condition $o%sli)*nflowB+(tflowvelocity

+(tflowBPress(re

(! !

v! v!

)! !

=hy do yo( have to s)ecify a bo(ndary condition for the o(tlet Remember that C+M&+ (sesa finite element method to obtain n(merical res(lts and is not solving for an e'(ation. *n this

regard the )ress(re derivative is an additional e'(ation and re'(ires an additional s)ecification.

#his s)ecification is to set the o(tlet )ress(re to ! Pa ga(ge.

Mesh Generation

&elect the mesh generation b(tton

"



Computing the Solution

Click the Sole b(tton in the Main toolbar.

Postprocessing and Visualization

#he defa(lt fig(re shows the velocity field, which is the absol(te val(e of the velocity vector. *nthis gra)h notice that the flow transitions from a (niform velocity at yD! and a f(ll develo)ed

velocity )rofile. 2etermine at what y distance the velocity )rofile can be considered f(lly

develo)ed. E/)lain yo(r answer.@dd arrows showing the magnit(de and direction of the velocity. Remember to select the @rrow

Plot o/.

:



?



;



*t is also )ossible to create cross%sectional

)lots of the velocity along hori<ontal lines atdifferent )ositions along the channel.

1. &elect Cross!Se+tion Plot Parametersin the Post/ro+essin) men(.

. &elect Line*E2trusion /lot as Plot

ty/e.3. Click the Line*E2trusion tab.

0. #he y!a2is data is by defa(lt set to

@elo+ity field.

>. &elect / as the /%a/is data from thedro) down men(.

". &et the Cross!se+tion line data

according to the table below8

PR+PER# @UE/! %1e%

/1 1e%

y! !

y1 !

:. &elect the Multi/le /arallel lines

check bo/.

?. &elect the @e+tor with distan+es

b(tton and ty)elinspace(0.00,0.05,9) in the

@e+tor with distan+es edit field. #he

lins)ace command will create an

e'(idistant vector from !.!! m to !.!>m with ; ste)s.

;. &elect ine &ettings and click on the

egend bo/1!. Click O7 .

#his gives a )lot in which velocity is on the y

a/is and the width of the )late is on the /%a/is.

Each c(rve starting with the bl(e linecorres)onds to a given distance in the flow

direction 4y direction5. $otice the a))earance

of the hori<ontal red lines on the original )ost )rocess conto(r sol(tion )lot shown above.

#hese lines are the ; cross sections that were

re'(ested.

1!



#he res(lting )lot shows how the velocity )rofile develo)s along the main direction of the flow.

@t the o(tlet, we can see that the flow

a))ears nearly to be f(lly develo)edand is a )arabolic velocity )rofile.

#o refine the sol(tion at the initial

bo(ndary, yo( can refine the mesh andthen solve again.

#he )lot below was )rod(ced byselecting the b(tton refine selection

and then dragging yo(r mo(se over the

mesh in the entrance region of the slot.

@lso if yo( wish to vis(ally com)are )lots yo( need to $otice how the

)roblem with the discontin(ity

red(ces.

11



Problem ?A Fully deelo/ed flow between ? /arallel /lates

$ow letGs look at f(lly develo)ed flow between )arallel )lates. C+M&+ has a bo(ndary

condition that allows (s to s)ecify the inlet )ress(re. For this )roblem what is the )ress(re dro) )er (nit length

#he sol(tion given in 9eanko)lis for this )roblem is ( )

!

1 x x

dy

dpv y −=

µ

=here / is half of the distance

between the )lates which in o(r geometry is e'(al to !.!1m. #he

ma/im(m velocity is given as

( )

!ma/

1 x

dy

dpv y

−= µ

. Using the

ma/im(m velocity of the f(lly

develo)ed flow from yo(r )revio(s

sim(lations and )hysical )ro)ertiesand dimensions calc(late the )ress(re

dro). Use this val(e of )ress(re to

s)ecify the inlet bo(ndary condition. Remember to convert )ress(re dro) into )ress(re.

1



Plot the /ressure /rofile usin) the surfa+e /lot /ressure and a Cross se+tion /lot.

E/)lain why there is a difference between the finite element method sol(tion 4C+M&+5 andthe analytical sol(tion.

13



Laminar Flow – Fallin) film

$ow solve a )roblem (sing a gravity term. *n this case yo( will sim(late a vertical )late with

li'(id flowing down the )late. Change yo(r geometry so that the film thickness is !.!!m andthe length of the )late is !.!1 m. &et the free film at / D ! and the wall at / D !.!! m. o( have

no a))lied )ress(re terms, b(t yo( will (se a force term. #he vol(me force vector, F D 4F x, F y, F

z 5, describes a distrib(ted force field s(ch as gravity. #he (nit of the vol(me force isforceBvol(me. *n this model yo( will need a force )er area term which o)erates in the negative

y%direction. @dd a gD;.?1 mBs to yo(r constants. Com)are yo(r sim(lation to an analyticalsol(tion that yo( obtain from first )rinci)les. &how this derivation.

Boundary Conditions

&elect Boundary Settin)s from the Physi+s men( and enter the bo(ndary conditions according

to the following table8

BO%N;&9< 5 ? 3 >

o(ndary condition &li)B&ymmetry +(tflowBPress(re $ormalflowBPress(re

$o%sli)

)! ! !

10



1>



SubmitA

5. ;eriations

1.1. of velocity )rofile for flow between )arallel )lates1.. of velocity )rofile for falling film

?. Cross Se+tion PlotsA

.1. 2evelo)ing flow between )arallel )lates.. F(lly 2evelo)ed flow between )arallel )lates

.3. Falling Film

3. Contour Plots 1Basi+ Solution6 @dd arrows on to) of )lot to signify velocity3.1. 2evelo)ing flow between )arallel )lates

3.. F(lly 2evelo)ed flow between )arallel )lates

3.3. Falling Film

>. E2+el PlotsA

0.1. com)arison of F(lly 2evelo)ed flow between )arallel )lates with analytical &ol(tion

0.. com)arison of falling film velocity )rofile with analytical sol(tion.

. &nswers

>.1. 2etermine at what y distance the velocity )rofile can be considered f(lly develo)ed forthe flow between )arallel )lates with (niform inlet velocity. E/)lain yo(r answer.

>.. )ress(re dro) calc(lation for f(lly develo)ed flow>.3. Comment on com)arison between analytical and sim(lated res(lts in the two f(lly

develo)ed flow models.

Fun for e2/erts and those who finish early

Laminar Flow Between Doriontal Plates with One Plate Moin)

$ow have the motion of the fl(id created by the movement of one )late. &et the bo(ndary

conditions for the entrance and e/it to give the f(lly develo)ed flow sol(tion 4 $ormalFlowBPress(re, with )ress(re e'(al to <ero5. o( donGt have to change the geometry. (st

e/cl(de gravity from any of the model terms. 4yo( already have done this5. Use the f(lly

develo)ed bo(ndary condition. Please note that yo( are solving a linear e'(ation.Plot8 Cross section )lot

1"

flow between parallel plates - femlabrev2006

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