materi 3 - flow of fluid through fixed beds

8/17/2019 Materi 3 - Flow of Fluid Through Fixed Beds

1/40

Flow of Fluids through

Fixed Beds (Granular andPacked Beds)

J. F. Richardson, J. H. Harker, J. R. Backhurst, Coulson and Richardsons Che!ical"ngineering, #olu!e $% Particle &echnolog' and earation Processes, edisi ke *, $++$


2/40

.-. /&R012C&0/

• &he 3ow of 3uids through 4eds co!osed ofstationar' granular articles is a fre5uentoccurrence in the che!ical industr' and

therefore exressions are needed to redictressure dro across 4eds due to theresistance caused 4' the resence of thearticles.

• For exa!le, in 6xed 4ed catal'tic reactors,

such as 0$708 con9erters, and dr'ingcolu!ns containing silica gel or !olecularsie9es, gases are assed through a 4ed ofarticles.


3/40

A. In streamline flow, there is only

velocity gradient on particle surface

(viscous drag=friction drag). Re1 !

". In tur#ulent flow, there is completetur#ulence in fluid (due to pressure

drag=form drag) $ minor role of

velocity gradient on particle surface

(viscous drag). Re1 % 1&&.

'. In transition flow, fluid is mied

laminar and tur#ulent. ! Re1 1&&

4.2. FLOW OF A SINGLE FLUIDTHROUGH A GRANULAR BED

Re1=

u1 ρ .

d m*µ =

uc*

e.e*(

S (1+

e)).ρ*µ


4/40

•4.2.1. Darcy’s law and !r"!a#$l$%y• &he 6rst exeri!ental work on the su4:ectwas carried out 4' 1arc'( - ) in -;8+ in 1i:on

when he exa!ined the rate of 3ow of waterfro! the local fountains through 4eds ofsand of 9arious thicknesses.

• t was shown that the a9erage 9elocit', as!easured o9er the whole area of the 4ed,

was directl' roortional to the dri9ingressure and in9ersel' roortional to thethickness of the 4ed.


5/40

• &his relation, often ter!ed 1arc's law,has su4se5uentl' 4een con6r!ed 4' anu!4er of workers and can 4e written asfollows%

• .

uc is #ased on #ed cross+sectional area, A


6/40

• &he linear relation 4etween the rate of 3owand the ressure di


7/40

•where μ is the 9iscosit' of the 3uid and B is ter!ed the er!ea4ilit' coe=cient forthe 4ed, and deends onl' on theroerties of the 4ed.

• &he 9alue of the er!ea4ilit' coe=cient isfre5uentl' used to gi9e an indication ofthe ease with which a 3uid will 3owthrough a 4ed of articles or a 6lter!ediu!.

•o!e 9alues of B for 9arious ackings,taken fro! "isenkla!($ ), are shown in

&a4le .- (for the la"$nar 3ow regi!e).


8/40


9/40

/ollow fi#re pac0ing


10/40


11/40


12/40

•4.2.2. S!c$&c s'r(ac! and )*$da+!

• &he general structure of a 4ed of articles

can often 4e characterised 4' the seci6csurface area of the 4ed SB and the

fractional 9oidage of the 4ed e.

•SB is the #!d s'r(ac! ar!a resented to

the 3uid er unit )*l'"! *( #!d whenthe articles are acked in a 4ed. ts unitsare (length)>-.


13/40

•e is the (rac%$*n of the 9olu!e of the 4ednot occuied 4' solid !aterial and ister!ed the fractional 9oidage, 9oidage, ororosit'. t is di!ensionless. &hus thefractional 9olu!e of the 4ed occuied 4'solid !aterial is ( - > e).

•S is the seci6c surface area of thearticles and is the s'r(ac! ar!a of aarticle di9ided 4' $%s )*l'"!. ts unitsare again (length)>-. For a shere, for

exa!le%


14/40

• t can 4e seen that S and SB are not e5ual due

to the 9oidage which is resent when thearticles are acked into a 4ed.

• f oint contact occurs 4etween articles sothat onl' a 9er' s!all fraction of surface areais lost 4' o9erlaing, then%

• ?s e is increased, 3ow through the 4ed4eco!es easier and so the er!ea4ilit'coe=cient B increases@ f the articles arerando!l' acked, then e should 4earoxi!atel' constant throughout the 4edand the resistance to 3ow the sa!e in all


15/40

•4.2.,. G!n!ral !-r!ss$*ns (*r *w%/r*'+/ #!ds $n %!r"s *( 0ar"an*3!ny !'a%$*ns

•Streamline fow (laminar fow)

• &he analog' 4etween strea!line 3owthrough a tu4e and strea!line 3owthrough the ores in a 4ed of articles isa useful starting oint for deri9ing ageneral exression.

• &he e5uation for strea!line 3ow througha circular tu4e is%


16/40

• f the free sace in the 4ed is assu!ed toconsist of a series of tortuous channels,e5uation .* !a' analogous for 3ow througha 4ed as%

u1 is #ased on e-uivalent diameter of pore channels

2o #e replaced #y

easily measured

varia#les


17/40

• t should 4e noted that ul and l' in e5uation.A now reresent conditions in the oresand ≠ uc and l in e5uations .- and .$.Howe9er, it is a reasona4le assu!tion that

l' is directl' roortional to l. 1uuit( )related uc and u- 4' the following argu!ent.

• n a cu4e of side X , the 9olu!e of freesace is eX 8 so that the !ean cross

sectional area for 3ow is the free 9olu!edi9ided 4' the height, or eX $ (o4ser9ed as$1 3ow). &he 9olu!e 3owrate through thiscu4e is uc X

$ (o4ser9ed as 3ow fro! outside),so that the a9erage linear 9elocit' through

the ores, ul, is gi9en 4'%


18/40

•?lthough e5uation . is reasona4l' truefor rando! ackings, it does not al' toall regular ackings.

•For e5uation .A to 4e generall' useful, anexression is needed for d'! , the

e5ui9alent dia!eter of the ore sace.

DoEen'( *,A ) roosed that d'! !a' 4etaken as%


19/40

• &hen taking u- uc /e and l' α l, e5uation.A 4eco!es%

•K'' is generall' known as DoEen's

constant and a co!!onl' acceted 9aluefor K '' is *.•?s will 4e shown later, howe9er, K'' isdeendent on orosit', article shae,

and other factors.-. .3 is #ased on parameters of outside

-. .3 is derived from -. ., #ut similar to -4s. .1 or .!

has used easily

measured

varia#les


20/40

•Co!arison with e5uation .$ shows thatB, er!ea4ilit' coe=cient, is gi9en 4'%

• nserting a 9alue of * for K'' in e5uation.%

•For sheres% S A /d and e5uation .--4eco!es


21/40

• .

•For nonsherical articles, the auter

!ean dia!eter ds should 4e used in lace

of d (see Chater -, e5uation -.-*)


22/40

Streamline and turbulent fow

• "5uation . alies to s%r!a"l$n! 3owconditions, though Car!an( ) and others ha9e

extended the analog' with ie 3ow to co9er4oth strea!line and tur4ulent 3ow conditionsthrough acked 4eds.

• n this treat!ent a !odi6ed friction factorR

-

/ρu-

$ is lotted against a !odi6ed Re'nolds

nu!4er Re-. &his is anal*+*'s to lotting

R/ρu$ against Re for 3ow through a ie as in#olu!e -, Chater 8.

• &he !odi6ed Re'nolds nu!4er Re- is o4tained4 takin the sa!e 9elocit and characteristic


23/40

• .

• &he friction factor, which is lotted against the!odi6ed Re'nolds nu!4er, is R- /ρu-

$, whereR- is the co!onent of the drag force er 'n$%

ar!a *( ar%$cl! s'r(ac! in the direction of!otion. R- can 4e related to the roerties of

the 4ed and ressure gradient as follows.• Considering the forces acting on the 3uid in a4ed of unit crosssectional area and thicknessl, therefore the total article surface area'n$%

crosssectional #!d ar!a SB.Ac.l /Ac ( - >

Re1 = u1 ρ .d m*µ = uc*e.e*(S (1+e)).ρ*µ

S (1+e) l = S B l = surface area of particles*volume of #ed thic0ness of #ed

#ased on outside(easily measured

varia#les)


24/40

• &hus the ar%$cl! area resistance force'n$% 4ed cr*ss5s!c%$*nal ar!a is R-Sl( - > e). &his

force on the 3uid force roduced 4' aressure di


25/40

• Car!an found that when R- /ρu-$ was lotted

against Re- using logarith!ic coordinates,

his data for the 3ow through rand*"ly

ac6!d #!ds *( s*l$d ar%$cl!s could 4ecorrelated aroxi!atel' 4' a single cur9e(c'r)! A, Figure .-), whose generale5uation is%

• Fro! e5uation .-A it can 4e seen that for

9alues of R!1 7 a#*'% 2, the second ter! is


26/40


27/40

• "5uation .-; can 4e o4tained fro! "5. .-*(force 4alance alica4le for 6xed 4eds) 4'su4stituting "5. .-- (for la!inar 3ow) for>∆P/l. &his gi9es%

6rom general force

#alance in fied #eds

6or laminar

flow


28/40

• ?s the 9alue of Re- increases fro! a4out $ to

-++, the second ter! in e5uation .-A4eco!es !ore signi6cant

• ?s Re- I -++ the lot is aroxi!atel' linear.

&he change fro! co!lete strea!line 3ow toco!lete tur4ulent 3ow is 9er' gradual

4ecause 3ow conditions are not the sa!e inall the ores.

• &hus, the 3ow starts to 4eco!e tur4ulent inthe larger ores (when e 4eco!es larger).


29/40

• t is ro4a4le that the *w n!)!r #!c*"!s

c*"l!%!ly %'r#'l!n% s$nc! s*"! *( %/!

assa+!s "ay #! s* s"all %/a%

s%r!a"l$n! c*nd$%$*ns r!)a$l !)!n a%

/$+/ *wra%!s.

• Rings, which as descri4ed later are often

used in industrial acked colu!ns, tend tode9iate fro! the generalised cur9e ? on

Figure .- articularl' at high 9alues of Re-.


30/40

• awistoski () co!ared the results o4tainedfor 3ow of 3uids through 4eds of /*ll*w r!ac6$n+s and has noted that e5uation .-A

gi9es a consistentl' low result for these!aterials.

• He roosed%

• &his e5uation is lotted as c'r)! B in Figure.-.


31/40

• For 3ow through r$n+ ac6$n+s are oftenused in industrial acked colu!ns, "rgun( -+ )o4tained a se!ie!irical correlation forressure dro

• riting d A /S (fro! e5uation .8)%

• &his e5uation is lotted as c'r)! 0 in Figure.-.

Compare Eq’s 4.16, 4.19, and 4.21 for different packings (random

granular packing, hollo packing and ring packing in !ig 4.1"#

6orce #alance


32/40

/ollow fi#re pac0ing


33/40

• &he for! of e5uation .$- is so!ewhat si!ilarto that of e5uations .-A and .-, in that the6rst ter! reresents )$sc*'s l*ss!s which are

!ost signi6cant at low 9elocities and thesecond ter! reresents 6$n!%$c !n!r+yl*ss!s which 4eco!e !ore signi6cant at high9elocities.

• &he e5uation is thus alica4le o9er a widerange of 9elocities and was found 4' "rgun tocorrelate exeri!ental data well for 9alues ofRe- /( - > e) fro! - to o9er $+++.


34/40

•E-a"l! 4.1

• &woheat sensiti9e organic li5uids, ofa9erage !olecular weight of -** kgk!ol,

are to 4e searated 4' 9acuu! distillationin a -++ !! dia!eter colu!n acked withA !! stoneware Raschig rings.

• ? condenser in the to of the colu!n, a still

for 4oiling the li5uids in the 4otto! of thecolu!n.

• &he ressure of 4oiler is ;+++ /!$ (Ps)

corresonding to 8*D (T s

) of 9aour

ressure.


35/40

• f the 9aour rate in colu!n is +.++ kgs,calculate the ressure in the condenser (Pc)

• t !a' 4e assu!ed that a SB S(-e)

;++ !$ / !8, μ +.+$ !/ s!$, e +.$ andthat te!erature changes and the correctionfor li5uid 3ow !a' 4e neglected.

7iameter=

1&&mm

height= !.m

8apour

rate =&.& 0g*s

condenser

"oiler (still)


36/40

6or Raschig ring pac0ing

general force #alance


37/40

general force #alance

for fied #ed

g

for fied #ed


38/40


39/40

199 = molecular weight

!!. = volume (m:) of 1 0mole at !;:< and 1 atm

1 0mol at standard condition

(&,2&) = !!. m:. s8s*2s =

&8&*2&8&*8s= 2&*2s s*&

>ass*8s=mass*8&2&*2ss*&ρs=(mass 10mol*8&)2&*2ss*&

2emp changes

neglected


40/40

materi 3 - flow of fluid through fixed beds

Documents