@uantitati2e *no ledge o. 6eha2iour o. .luids allo s us todesign, optimise, and anal!se industrial 6ioprocessings!stems hich in2ol2e .luids Since almost all 6iochemicalreactions re5uire a5ueous media in order to proceed, thismeans that .luid mechanics can pla! a 6ig part in large scale6iochemical processing operations Particular areas o.importance are%
• .lo regimes and mi ing e..icienc! in 6ioreactors• scale3up o. aero6ic .ermentations
• e..icienc! and throughput in 6ioseparation units• pumping and .lo measurement o. 6ioprocess .luids
9i5 speci.ic gra2it! atio o. li5uid densit! to that o.pure ater at Cstandard$
temperature
32 P$e!!u$e >ll .luids e ert a .orce normal to a solid 6oundar! or an!plane dra n through the .luid This is due to the multitude o.collisions 6et een the .luid molecules and the solid6oundar! or plane This .orce, e pressed per unit area o. the6oundar! or plane, is termed the pressure
Pascal’s Law states% Assuming that there is no relativemotion beween adjacent layers of fluid (no shear forces),the pressure at any point in the fluid is the same in alldirections.
37 a#&u$ P$e!!u$e
>t an! li5uid/gas inter.ace there is a continuous interchangeo. li5uid molecules across the phase 6oundar! "hen the
2olume o. the gas phase is restricted and assuming thatthere is su..icient li5uid present, this interchange e2entuall!reaches a stead! state, in hich the rate o. e2aporation isthe same as the rate o. condensation Dnder suchcircumstances, the gas phase 6ecomes saturated ith theli5uid 2apour Thus the partial pressure o. the li5uid 2apour
in the gas phase is constant, and is termed the saturationvapour pressure
For a gi2en li5uid the saturation 2apour pressure ill 6edependent onl! on temperature This is due the .act that astemperature increases, so too does the 2elocit! o. the li5uidmolecules, hence allo ing them to enter the gas phasemore readil! Thus saturation 2apour pressure increasesith temperature
100 101'00 $altons Law of Partial Pressures 5uanti.ies the pressure6eha2iour o. gas/2apour mi tures For e ample, in the caseo. gas containing m di..erent gaseous components, andhich is in e5uili6rium ith a li5uid, the total pressure, P Total ,o. the gas phase is gi2en 6!%
P p pTotal x gas component vapour
x
x m
= +=
=
∑1
(1 1)
34 C&+#$e!!i,ili%(
The compressi6ilit!, or change in 2olume due to change inpressure, o. .luids 2aries dramaticall! on going .rom li5uidsto gases Since the molecules in a li5uid are much closertogether than those in a gas, it might 6e e pected that the.ormer ould 6e unli*el! to 6e pushed much closer togetheron increasing the e ternal pressure on the li5uid This is6ecause the electron clouds on adEacent li5uid molecules
ould electrostaticall! repel one another ases, on theother hand are much less dense, and thus on increasing thee ternal pressure, there is much more scope .or pushing thegas molecules into a smaller 2olume
Ta,le 32 Fluid!9 Cha 'e i &lu+e /&$ a T5&/&ldI c$ea!e i *##lied P$e!!u$e
%luid &#olume $ecrease"ater 0 00#'
deal as '0 00
The data in ta6le 1 8 illustrate the .act that gases are
appro imatel! a .actor o. 10#
more compressi6le than aterThus the normal assumption is that li'uids areincompressible hereas gases are compressible ( t should6e noted that these assumptions are not correct in certainsituations, such as the C ater hammer$ phenomenon or ithgases at 2er! high pressure)
;ompressi6ilit! is normall! important in dealing ithsituations in hich the .luid in 5uestion is a gas
3: i!c&!i%(
The 2iscosit! o. a .luid is a measure o. its resistance toshear or angular de.ormation Thus as a .luid is de.ormed6ecause o. .lo or applied e ternal .orces, .rictional e..ectsoccur due to the motion o. the .luid molecules relati2e toeach other Such e..ects strongl! in.luence the d!namic
6eha2iour o. .luids (i e during .lo )n order to o6tain a more 5uantitati2e de.inition o. 2iscosit!,consider a thin la!er o. .luid 6et een t o parallel planes,distance d! apart, as sho n in .igure 1 1 The lo er plane is.i ed and a shearing .orce, F is applied to the upper plane,
causing it to mo2e in the direction at a constant 2elocit!,du
F Gelocit! du
d! 9inear 2elocit! pro.ile!
Stationar!, area >
Fi'u$e 3 el&ci%( '$adie % a d !hea$ !%$e!! i a /luid
Particles o. the .luid in contact ith each plate ill adhere toit, and i. the distance d! is not too great, or the 2elocit! dutoo high, the 2elocit! gradient or pro.ile ithin the .luid la!erill 6e linear Thus the .luid 6eha2es as i. it comprised a
series o. thin sheets, each o. hich ould slip a little relati2eto the ne t Dnder stead! state conditions the .orce F ill 6e6alanced 6! an internal .orce in the .luid due to its 2iscosit!The shear .orce per unit area (or shear stress, ! ) isproportional to the 2elocit! gradient (or shear rate) in the.luid%
F A
R du
dy y x= = µ ewton’s 'uation of #iscosity (1 8)
Transposing this e5uation, e ha2e%
µ =
R
du dy y
x(1 &)
The proportionalit! constant, µ, is termed the 2iscosit!,2iscosit! coe..icient, or a6solute 2iscosit! o. the .luid< pressing e5uation 1 & in ords, e can sa! that the
a6solute 2iscosit! is e5ual to the shear stress di2ided 6! theshear rate > ewtonian .luid is one in hich the 2iscosit!coe..icient remains constant o2er a ide range o. shearrates, assuming constant temperature and pressure non* ewtonian .luids, µ can 2ar! ith shear rate in anum6er o. a!s, depending on the .luid t!pe
+inematic viscosity, ν , o. a .luid is de.ined as the ratio o. the2iscosit! coe..icient to the .luid densit!, ρ%
ν µ
ρ = (1 #)
The 2iscous or .riction .orces in .luids are caused 6! 6othcohesion (intermolecular attractions), and momentuminterchange 6et een molecules in adEacent la!ers in the.luid As temperature increases, the viscosities of all li'uidsdecrease, whilst those of gases increase This is 6ecausethe (relati2el! short range) cohesi2e .orce, hich decreasesith increasing temperature, is the maEor contri6ution to
2iscosit! in li5uids n the other hand .or gases, thepredominating .actor is the (long range) interchange o.molecules 6et een the la!ers o. di..erent 2elocities Thus amolecule hich di..uses .rom a rapidl! mo2ing la!er into aslo er mo2ing la!er, ill tend to speed up the molecules inthe latter, and 2ice 2ersa This molecular interchangeproduces a .riction .orce 6et een adEacent la!ers Hence,since molecular di..usion increases ith temperature (*inetic
theor! o. gases), the 2iscosit! o. the gas ill also increaseThe a6solute 2iscosit! o. all .luids is normall! independent o.pressure, e cept at e tremel! high gas pressures ( here the*inetic theor! o. gases no longer applies) The *inematic
2iscosit! o. gases, ho e2er, does 2ar! ith pressure, due topressure3induced changes in gas densit!
36 Su$/ace Te !i& a d Ca#illa$(
>lthough the molecules ithin the 6od! o. a li5uid areattracted e5uall! in all directions 6! the surroundingmolecules, at the li5uid sur.ace the molecules are onl!attracted in ards and do n ards 6! the molecules 6eloThis e..ect causes the li5uid sur.ace to 6eha2e as i. it erean elastic mem6rane under tension he surface tension, σ ,is defined as the force acting per unit length on a line drawn
on the li'uid surface Sur.ace tension is constant at an!temperature .or the sur.ace 6et een an! t o particularphases (li5uid3gas or li5uid3li5uid), and it decreases ithincreasing temperature
The net e..ect o. sur.ace tension is to reduce the sur.ace o.a .ree 6od! o. li5uid to a minimum, since to ma*e the li5uidsur.ace area larger ould re5uire 6ringing 6ul* molecules tothe sur.ace against the un6alanced in ard attraction o. themolecules 6elo Thus drops o. li5uid tend to ta*e aspherical in order to minimise sur.ace area n smalldroplets, sur.ace tension ill cause an increase in internalpressure, p, in order to 6alance the sur.ace tension .orce;onsider the .orces acting on a diametral plane through aspherical drop o. radius r%
Force due to internal pressure p πr 8 Force due to sur.ace tension at the perimeter 8 πr σ
For small 6u66les in a li5uid, i. this (sur.ace tension3induced) pressure is greater than the pressure o. gas or2apour in the 6u66le, the 6u66le ill collapse Sur.acetension e..ects are thus particularl! important in 6ioprocessengineering situations here it is desired to e..icientl!dissol2e up a gas in a li5uid, i e in aero6ic .ermentations
The capillary effect is due to 6oth cohesi2e and adhesi2e.orces The .ormer .orce results .rom the intermolecularattractions ithin the li5uid, hereas the latter is due toattractions 6et een the li5uid and a solid sur.ace ith hichit is in contact "hen cohesion is o. less e..ect than
adhesion, then the li5uid ill et the solid and spread or riseat the point o. contact . the cohesi2e .orce predominates,then the li5uid sur.ace ill 6e depressed 6elo its true le2eland ill see* to minimise contact ith the solid Thus aterill sho a capillar! rise in a narro glass tu6e hereas
mercur! sho s a capillar! depression
The capillar! e..ect ma! 6e 5uanti.ied 6! e5uating the .orcesacting on the column o. li5uid in a narro tu6e, as sho n in.igure 1 8
8r θ
h
Fi'u$e 32 Ca#illa$( $i!e
Dp ard .orce due to sur.ace tension/ etting 8 πr σ cos θ Ao n ard .orce due to eight o. li5uid column πr 8h ρg
here θ contact ( etting) angleThus the capillar! rise (or .all) is gi2en 6!%
h gr =
2σ θ ρ
cos(1 -)
;apillar! e..ects are important in situations here li5uids arecontained in narro tu6es The capillar! e..ect in tu6es o.diameter greater than ca 1 8cm is usuall! negligi6le
>t the outset e can de.ine .luid statics as the study offluids at rest or under situations where there are no shearingforces acting on the fluid (i e no relati2e motion 6et eenadEacent la!ers) The latter part o. this de.inition ould thusinclude situations such as .luids in mo2ing containers ngeneral, this aspect o. .luid mechanics deals ith the.ollo ing% prediction and measurement o. li5uid pressureand h!draulic head, 2ariation o. gas pressure ith altitude,and 6uo!anc! o. solids nl! the .ormer o. these is o.immediate concern in this course
23 a$ia%i& &/ P$e!!u$e i a Fluid ; de$ G$a i%(
;onsider an element o. .luid, o. mass m, consisting o. a2ertical column o. cross3sectional area >, sho n in .igure8 1%
n a .luid o. constant densit!, e5uation 8 8 can 6e integrateddirectl! to gi2e%
p 3 ρgI ? constant (8 &)
(7ote% this is a general .orm o. e5uation 8 1)
n a li5uid, the depth I is normall! measured downwards.rom the .ree sur.ace so that I 1 3h and I 8 0 Thus
e5uation 8 1 6ecomes%
p1 ρgh ? p 8
n addition, i. it is assumed that the .ree sur.ace o. the li5uidis at atmospheric pressure, p atm %
p a6s ρgh ? p atm (8 #)
This e5uation gi2es the a6solute pressure at depth h Thea6solute pressure scale ta*es Iero pressure as per.ect2acuum n engineering it is o.ten more normal to e presspressure in terms o. gauge pressure, here Iero pressure ista*en as 1 atmosphere, i e p gauge p a6s 3 patm Thus 8 #6ecomes%
pgauge ρgh (8 ')
<5uation 8 ' sho s that the gauge pressure at an! point ina li5uid is de.ined 6! li5uid densit! and the 2ertical height (orhead ) o. that point .rom the sur.ace t should 6e noted thathere pressures are 5uoted as head, it is important that
either the .luid is named or its densit! is 5uoted, e g - 'm o.ater or '#0mm o. mercur!
>n important outcome o. e5uation 8 ' is that the pressuree erted by a fluid is not affected by the weight of fluid
present, only by the vertical li'uid depth and its density. The.ollo ing t o 2essels ha2e 6een .illed to the same depthith li5uids o. the same densit!%
h
>rea >
Fi'u$e 232 The h(d$&!%a%ic #a$ad&@
Pressure on 6ottom in each case, p ρghForce on 6ottom in each case p> ρgh>
Thus, although the eight o. li5uid in 6oth 2essels iso62iousl! di..erent, the .orce on the 6ases o. the 2essels isthe same
n terms o. their d!namic properties, .luids ma! 6e classi.iedin t o a!s%
A!namic 6eha2iour o. .luids
eaction to e ternall! eaction to shear stressapplied pressure
mportant .or compressible mportant .or all .luids.luids (gases)
Gelocit! gradients caused 6! 2iscous .orces in 6oth li5uidsand gases, determine .lo 6eha2iour and result in thedissipation o. energ! ith the mo2ing .luid >ll gases andmost pure li5uids e hi6it 7e tonian 6eha2iour, hereassome li5uids, particularl! those hich contain a secondphase in suspension, sho non37e tonian 6eha2iour
73 Fluid Fl&59 Fu da+e %al C& ce#%!
"hen an! .luid .lo s in contact ith a solid, e g in a pipe oro2er a sur.ace, .luid 2iscosit! causes non3uni.orm 2elocit! ina plane at right angles to the .lo direction This 2ariation in2elocit! can 6e represented diagrammaticall! 6! the use o.streamlines dra n in the .luid A streamline is defined as animaginary line in the fluid, across which (i.e. at right anglesto which) at any given instant, there is no macroscopic flow.Thus the 2elocit! direction (or 2ector) o. the .luid particles in
the streamline is al a!s along the line The .lo rate6et een an! t o adEacent streamlines is al a!s constantHence constant .luid 2elocit! o2er a cross3section isindicated 6! e5uidistant streamlines, and i. the! are pac*edcloser together then 2elocit! is increased o2er the cross3section The 6oundaries o. the .lo (e g the internal all o.a pipe) are al a!s included as streamlines, since no .locan occur across them
Streamlines tell us a6out%• direction o. .luid 2elocit! ithin the .lo (arro s and line
shapes)
• i. there is non3uni.ormit! in .luid 2elocit! across the .locross3section (non3e5uidistant spacing)• i. there is 2ariation in .luid 2elocit! along the direction o.
Ae.ine stead! .lo % all conditions at an! point sta! constantr t time
;ontri6utions to energ! in a .luid%
1 nternal energ! (D)8 Potential energ! (Ig, here I height a6o2e datum)& Kinetic energ! (u 8/8, here u 2elocit!)# Pressure energ! (P2, here 2 2olume/unit mass o. .luid)
>ssume% stead! .lo , and .luid o. constant densit! >ppl! principle o. conser2ation, e g 1 st 9a o. TdLs, to .luidmo2ing in a single streamline%
)(2
2
Pvu
z g U W q s ∆+∆+∆+∆=+ (& 1)-eneral 'uation for nergy of a %luid in otion
here% 5 net heat a6sor6ed .rom the surroundings " s net amount o. or* done 6! the surroundings(sha.t or*)
e riting (& 1) 6! considering conditions at t o points in thestreamline%
)(2
)()()( 1122
2
1
2
21212 v P v P
uu z z g U U W q s −+−+−+−=+
(& 8)
>ssume%1 That the .luid is incompressi6le (li5uid, or gas ith small
pressure drop along the streamline) Thus 2 1 2 8 const
8 7o sha.t or* done 6! the surroundings on the .luid& That the .rictional (2iscous) .orces caused 6! .luid motion
result in heat generation, and that this together ith an!heat a6sor6ed .rom the surroundings, results in a changein the internal energ! o. the .luid Thus 5 D 8 3 D1
This gi2es
)(2
)()(0 12
2
1
2
212 P P v
uu z z g −+−+−= ,
or, since 2 1/ ρ .or an incompressible fluid , and rearranging%
constant22
2
11
1
2
22
2 =++=++ u gz
P u gz
P ρ ρ
(& &)
/ernouilli0s 'uation1 relates the pressure at a point in afluid to its position (elevation) and velocity
Mo2ing .luid particles possess momentum . .luid 2elocit!changes in either magnitude or direction, there ill 6e acorresponding momentum change From 7e tonLs 8 nd 9a ,e can 5uanti.! the .orce hich ill 6e re5uired to producethis change%
Force rate o. change o. momentum m u/t (& ')
;onsider 83d .lo 6et een >B and ;A%
u 8
; φ
A
> !
u 1
θ
B
>ppl!ing & ' in the direction%
F (Mass/unit time) (change in 2elocit! in direction)
here u 8 and u 1 are the components o. u 8 and u 1
Similarl!%
F ! m(u 8 sin φ 3 u1 sin θ) m(u !8 3 u!1 )
These components can 6e com6ined to gi2e the resultant.orce%
F √(F 8 ? F !8)
The .orce e erted 6! the .luid ill 6e e5ual and opposite
Same approach applies to .lo s in &3d
n general .luid omentum 'uation states%
Total .orce e erted on ate o. change o. momentumthe .luid in a control in the gi2en direction o. 2olume in a gi2en the .luid passing throughdirection the control 2olume
F m(u out 3 uin) (& -)
73 37 Fl&5 C& %i ui%(
>pplication o. the principle o. mass conser2ation, to the .loo. .luid in a streamtu6e o. 2ar!ing cross3sectional area,gi2es the %low !ontinuity 'uation %
u1 > 1 u 8 > 8 , (& )
here > 1 and > 8 are the cross3sectional areas at theentrance and e it o. the control 2olume732 La+i a$ a d Tu$,ule % Fl&5
"hen the motion o. .luid particle in a stream is distur6ed(relati2e to the o2erall .lo direction), the generated inertiaill tend to cause the particle to continue mo2ing in the nedirection n the other hand the 2iscous .orces due to thesurrounding .luid ill tend to ma*e the particle con.orm tothe motion o. the rest o. the stream n streamline (2iscous).lo the 2iscous shear stresses are strong enough toeliminate the e..ects o. an! de2iation, 6ut in tur6ulent .lothe inertial .orces predominate, and hence large scalemo2ement out o. the direction o. .lo occurs
2atio of inertial1viscous forces is critical in determining flowtype.
7323 The Re( &ld! Nu+,e$
;onsider .luid .lo ing in a pipe Suppose 9 is acharacteristic distance o. the pipe, sa! its diameter, and t isa characteristic time For a small element o. .luid o. densit!ρ, e can sa!%
<lement 2olume * 1 9&
<lement mass * 1ρ9&
<lement 2elocit!, u * 89/t<lement acceleration * &9/t 8 , here * 1, * 8, and * & are constants 7e tonLs 8 nd 9a %
The 2alue o. the 5uantit! ρu9/ µ is thus the criterion hichdetermines hether a .lo is streamline or tur6ulent Since itis ratio o. .orces it is dimensionless t ma! also 6e rittenas u9/ ν, here ν is the *inematic 2iscosit! ( µ/ρ)
2eynolds umber (2 e ) ρu9/ µ u9/ ν (& +)
For .lo in pipes%
e Nca 8000 gi2es laminar .lo
e ca 8000 3 #000 gi2es transitional .lo (laminar ↔ tur6ulent c!cling)
Aue to the 2iscous and inertial .orces acting on a .luid as it.lo s through a pipe, it is ine2ita6le that energ! lossesoccur These mani.est themsel2es as a drop in pressure ora h!draulic head loss due to L.rictionL Such e..ects arerelated to a num6er o. .actors, including e!nolds 7um6erand the internal sur.ace roughness o. the pipe t is o.tenimportant that this t!pe o. loss 6e 5uanti.ied in order toassess, .or e ample, pumping re5uirements Pipe .rictioncorrelation charts are used to this e..ect
(See handout)
737 N& ?Ne5%& ia Fluid!
Pre2ious discussions% concerned ith 5uanti.!ing thed!namic 6eha2iour o. .luids hose 2iscosit! is notdependent on shear rate (i e 7e tonian .luids)
< pressing the generalised 737 6eha2iour 5uantitati2el!.rom e5uation 1 &%
=
dydu
R x y 1 (& 10)
==dydu
dydu
R x
x
ya 2 µ (& 11)
7o single parameter model .or the .unctions . 1 and . 8 illdescri6e 737 6eha2iour Garious multi3parameter modelsare used, and these can onl! 6e applied o2er a limited rangeo. shear rates .or a gi2en rheolog! t!pe
here !o is the !ield stress and µp is the plastic 2iscosit!
73732 Ti+e e#e de % Beha i&u$
n general .or shear thinning (pseudoplastic) .luids, µa
decreases ith time .or a step increase in shear rate This istermed thi otropy The opposite e..ect is o6ser2ed ithshear thic*ening .luids, and is termed rheope y H!steresisis o6ser2ed i. the shear rate is step increased and then stepdecreased
73737 i!c&ela!%ic Fluid!
Ha2e some o. the properties o. 6oth a solid (elastic) and ali5uid (creep/.lo ) Beha2iour o.ten 2er! comple Somee amples%
1 Stirred li5uid in a c!lindrical 2essel, ith s irling o. .luid indirection o. stirring% on cessation o. stirring, .luid starts tos irl in the opposite direction, i e it Lun indsL
8 n emerging .rom a tu6e, ma! .orm a Eet hich is largerthan the diameter o. the tu6e aperture (Ldie3s ellL)
& "eissen6erg <..ect%
7e tonian 737 Pseudoplastic 737 Giscoelastic
73734 N?N Fl&5 i Pi#e!
The .ollo ing relationships ma! 6e used to calculatepressure drop, 3 ∆P, or mean (linear) .lo 2elocit!, u,associated ith 737 .luid .lo in pipes%
here # o9/(3∆P d), and the other terms are as de.inedpre2iousl!
7ote in the case o. e5uation & 1-, more than one 2alue o.3∆P is possi6le .or a gi2en 2alue o. mean linear 2elocit!
urbulent flow of * fluids1
>t present there is no relia6le method .or predictingpressure drop in this situation Ho e2er, it should 6e notedthat 737 characteristics ha2e a much stronger in.luence on.lo in the streamline .lo regime, here the 2iscous
.orces are o. prime importance Dse o. the standard pipe.riction chart .or shear thinning .luids tends to o2er3predictpressure drop >lso laminar .lo can persist to slightl!higher e!noldLs num6ers in the case o. 737 .luids Thus ino2erall terms there is an in36uilt sa.et! .actor associated intreating 737 .luids as 7e tonian, in situations here the.lo is e pected to 6e tur6ulent
734 LiAuid Mi@i ';an .ul.il t o 6asic .unctions%• achie2e a predetermined le2el o. homogeneit!• impro2e heat trans.er The attainment o. mi ing in a li5uid ine2ita6l! re5uiresenerg! (in the .orm o. mechanical po er) and this ise2entuall! dissipated as heat 6! the li5uid
3 Mi@i ' Mecha i!+!
olecular diffusion Dltimatel! mi ing occurs at themicroscopic le2el due to molecular di..usion This isnormall! .airl! rapid compared to macroscopic mechanisms,
6ut it can 6ecome important, i e slo , in li5uids o. high2iscosit!
>dditional mi ing mechanisms in the laminar mi ing regime%
hinning 3 elongation of fluid elements due to laminar shearflow. Gelocit! gradients induced 6! laminar shear .orce hasthe net e..ect o. distorting .luid elements 6! thinning andelongating them%
Physical splicing. Fluid elements ma! 6e cut and mo2edrelati2e to one another to induce mi ing%
;ut
T ist
n the tur6ulent .lo regime, other more comple and lessunderstood mi ing mechanisms occur These primaril!in2ol2e tur6ulent edd! di..usion
23 Scale?;# &/ S%i$$ed e!!el!
The principles o. geometric and d!namic similarit! areapplied .or e..ecti2e scale3up The single most importantoperating parameter hich can maintain geometric andd!namic similarit! is impeller po er input, P ;omparison o.po er input magnitudes in di..erent siIed 2essels is.acilitated 6! use o. the po er num6er, 7 p%
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