Transitional Flow Phenomena in Concentric Annuli Richard W. Hanks' and William F. Bonner2 Department of Chemical Engineering, Brighani I'oung Cniversity, t'rouo, Ctah S.$SOl
An annular transition model wherein the flow between the core and the radius of maximum velocity ( r n t ) transists accompanied b y simultaneous stabilization of laminar flow between r m and the outer wall accounts for the known transition flow properties of annuli. During transition, r m deviates from its laminar value, through a maximum, returning to a high-Reynolds-number asymptote (less than the laminar value). A first critical Reynolds number delineates the core transition and a second indicates the outer wall transition. The second critical Reynolds numbers exceed the values expected if no core transition occurs first, showing the stabilizing effect of core transition on the outer flow region. Minor core eccentricities, which negligibly effect laminar or turbulent friction losses and velocity profiles, have a marked influence on these characteristics during transition, making quantitative prediction of transition phenomena difficult in practical systems.
T h e concentric annulus is a widely used flow geometry in t'he cheniical process industries and has been the subject of a considerable amount of investigation, both theoretically and experimentally, for many years. In addition to being of great practical importance, the annular geometry provides a unique testing ground for fundamental invest'igations of the phenomena of fluid mechanics. The work described in this paper was performed i n an at'tempt to discover the fundamental causes of the interesting transitional flow phenomena reported by Rothfus and coworkers in their studies of concent,ric nnniili (Croop, 1958; Crool) and Rothfus, 1962; Rothfus, 1948; Rothfus et al., 1950, 1958; Walker, 1957). 13cfore one undert'akes a detailed analysis of the transitional flow phenomena which ocsur in a concentric annulus, it is helpful to review certain key experimental observations from the work of previous investigators, in order to clarify the nature of the problems suggested thereby. In a series of velocity profile studies m i d e in t'raiisitional flow i n ctoncent'ric annuli, Rothfus and his coworkers (Croop, 1958; Croop antl Rothfus, 1962; Rothfus, 1948; Walker, 1957) pronounced skewing toward the inner boundary or core of the atinulus a1)parently was observed as transition began. This skewing of the profile is illustrated in Figure 1 for a set of data obtained by Rothfus (1948) in comparison with the theoretical profile (solid curve) corresponding to the same bulk flow Reynolds number. The point of special interest in Figure 1 is that the radius of maximum velocity, T,,, of the experimental curve is snialler than the corresponding radius, ,F for the t'heoretical laminar profile. Furthermore, the corresponding velocit'ies a t these locations are such t h a t v,, > fin,, and the portion of t'he data lying above and to the core side (left in the figure) of the theoretical curve is more blunted than the theoretical profile. These observations have been interlireted (Croo~)and Rot,hfus, 1962; Rothfus, 1948) as representing transition to turbulence in a localized region near the core prior to the onset of full turbulence throughout the entire flow field. X similar argument was proposed much earlier by Lea and Tadros (1931) to account for "a1)iiormd" pressure loss measurements in annuli as the transition to turbulence was approached from laminar flolv.
In addition to these velocit,y 11rofile observtitioiiu, frirtiori losses (Croop and Rothfus, 1962) were presented and discussed which indicated abnornial increases in drag 011 the ('art: associated with the inward shift of rrrle.However, :ilthouph they tentatively huggestetl n inechani>m for this l)eh:ivior, these investigators left the matter uiiresolved, i i i hope that later work might clarify the situation. Other studies of velocity profiles i l l high Rrytioltls nririi1)cr turbuleiit flow (15righton :tnd Joties, 1964; C ' l i i t i i l ) :tiit1 Ii\vusnoski, 1968; Lorenz, 1932; Mncagno atid l I ( * l ) o u g a l l ~1966; Ratkoiwky, 1966; JVilaon :md lletlwcll, 19671, both theoretical and experirueiitnl, have show^ coiiclusively that r,,!, (turhulent) < ? r , l . This ohservatioli coiitixtlicats t h r ronibined early results of Rothfus and his eoworkcrs, whicli iiitlicated that r,, (turbulent) = ?,,l. 111 a later > t i d y , ltotliius e t :i1. (1966) 1)erforrnecl further studies;, ii>iiig :t helit I'itot tube rather than the side-ol)ening inipact tube used in the earlier work. They werc able to clulilicate thv high Itt~yiiolds number result of Brighton and Jones (1964). I11 this paiier, however, liothfus e t al. :ilso claimed that for the lower rnnge of Reynolds numbers (800 to 10,000) in ivliich t h r transition occurs, their r,,, values esseiitiully r1ul)licntrcl those oMaiiied earlier by Walker (1957) : ~ n dIiy Croop u i d 11otlifu.s (1962). However, Croop antl liothfus (1062) had iiidic~iitrdt'he erroneous result that, r,, (turlnilent) = P,,,. 'This raises the question: Since the turl)uleiit limits of these 'r,,, data have been cleuly shown to be in error>:ire the traiisitiolial rnnge T,,,, data valid or are they also open to ciue>tion? K h a t behavior should occur as transition to tuhulence occurs in :L coiicentric :innulusl 1he above series of experimental observ:ition>, conflicting proposed explan:itions, and the questions r:iisetl thereby suggest that a detailed theoreticnl analysis of the I)lienoniena of t,ransition from laminar to turbulent flow ill a cwiretitiic annulus is needed. Only in this n1:iiiiier c:in it truly f u i i c h nieiital uiiderstaiiding of tliesc transitioii I)hciionieiia i n 1 ) : ~ ticular, xiid the phenoniena of traii>iti(iiii n general, l)cl Ilad. Such a n analysis, hased on the rutatioiial nionientuni 1)alanre of Hanks (1969), is the siil)ject theory of laniinar f l o stability ~ of the reniaiiider of the 1)reseiit 1):iper. 7 ,
Theoretical Anaiysis
To whom correspondence should be sent. 2 Present address, Battelle Northwest Laboratories, Richland, Wash.
Since the primary iiitercst of the present iiivcstigation is in the stability of laminar flow in a concentric annulus, the Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
105
,-"I-t
I
I
I
I
I
I
I
x
Continuity of velocity a t 4 = requires u l ( x ) = uo(x),which leads to the familiar expression for (Bird et al., 1960)
I
'Of
x
I n terms of the over-all friction factor, f = r2(l - u 2 ) ( - d p / d z ) / p ( ~ , ) ~and , the Reynolds number based on the hydraulic diameter, Re = 2r2(1 - g ) ( v , ) p / p , the coefficients of Equations 2 and 3 become
-
tc 01
0
03
02
04
05
E=
06
07
00
09
r/r2
Figure 1. Data of Rothfus (1948) showing skewing of profile toward inner boundary
- Theoretical laminar
profile
0 Actual reported d a t a Re = 1 8 2 0 , u = 0.162, X = 0 . 5 1 7 2 rme i, and vmg V,
where = r/r2, Pz = 1/2r2(1- X 2 ) ( d p / d z ) , = ?,jr2, and r 2 is the radius of the outer boundary of the annulus. Clearly = X corresponds from Equation 1 to the condition P,, = 0. Thus, also represents in dimensionless form the radius of zero shear stress. I n the following treatment we use a bar over h to designate its theoretical laminar value as distinguished from its value for any other condition which is represented simply as A. Because of this, i t folloivs from Equation 1 that P,, > 0 for [ < 5, corresponding to the fluid resisting a flux of axial momentum in the negat'ive r direction toJvard the core boundary which is located a t r = r I = orI. Similarly, P,, < 0 for X < 4 , correspondiiig to the fluid resisting a flux of axial momentum in the positive r direction toward the outer boundary which is located a t r = rI. We designate the region u 5 4 5 X as the inner region and the region X 5 E 5 1 as the outer region and refer to these regions accordingly in what follows. The solution of the equations of motion, originally given by Lamb (1932) can be expressed for these two regions as
x
where u = u,/(u,), (0,) is the aiea mean velocity of flow, and subscripts i and o designate inner and outer, respectively. 106 Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
Xz)
fRe
4(1 -
(5)
u ) ~
Actually, because of the definitions used for f and Re, Equation 5 is valid regardless of the flow regime in question. I n analyzing the transitional flow phenomena, this is a desirable feature, since no problem concerning the nature of h arises. I n the particular case of laminar flow it is easily shown that fRe
logical starting place is the equations of laminar velocity distribution. These equations are familiar from elementary texts on fluid mechanics; but in the present analysis we require them in a n equivalent, but rather unfamiliar, form. Therefore, it is essential to the later development that the appropriate form of the laminar flow equations be first derived here. Laminar Flow. T h e essential key in the analysis of the transition phenomena that occur in the concentric annulus is the division of the flow field into two separate zones, separated from each other b y the radius of maximum velocity, r,. The shear stress distribution, P,,, is well known to be (Bird et al., 1960)
-
r2P2
p(~J(1-
4(1
-
-
(1 -
62)
1/4(1 - u4) - X2(1 -
-
u ~ )
X4 In u
(6)
x
with being given by Equation 4. Flow Stability. FIRSTT R a s s I ~ r o s . Thus far in the analysis we have observed the existence of two regions of the flow field, each having a different velocity distribution and each being subjected to a different shear stress distribution. T h a t the latter condition IS true is easily shown. Consider the positions 5 = h 6 and E = h - 6, where 6 5 max (1 - h , h - u)-that is, two positions symmetrically spaced relative to the surface of zero shear stress E = A . From Equation l i t follows that
+
T h a t is, a t symmetrically spaced locations relative to 6 = A , the shear stress in the inner region exceeds in magnitude that in the outer region. Furthermore, if the same ratio of stress is taken a t the two boundaries (representing the maximal values of each stress), one finds for laminar flow
Clearly, then, the magnitude of shear stress distribution in the inlier region exceeds that in t'he outer region. This is just a reflection of t'he fundamental skewness of P,, in t,his system with respect to r. This is a spatial skewness, while the stress retains its tensorial symmetry, in that P,, = Pz,. Since this spatial stress asymmetry exists in the two regions of flow, each must be analyzed separately for its stability characteristics. The necessity for this bifurcation of the flow fields from the point of view of stability was not recognized in a n earlier treatment (Hanks, 1963). The fact t h a t the inner region is subjected to a higher shear stress field than the outer region suggests that it may reach the conditions of instability first. If one applies the rotational momentum balance stability theory (Hanks, 1969) to the inner region, he obtains the result
(fx)
where is given by Equation 6 and is given by Equation 4. The present system is rectilinear, for which case the
special form (Hanks, 1969) of the stability parameter applies. I n this situation the function F z ( t , u ) comes from calculating u t d u i / d t from Equation 2. When K(Eac) = 404 ( E t c is t h e value of t for which d K / d t = 0 and K is maximized), R e = Retc. T h e value of ttcis the root of the equation
dK d4
O = 3(ttc)4- [3X2 X2[(tiC)*
+
+ - X* In + X*(2X* - u* +- X* 111 u2](tic)2
62
x 2 1 111 ( t i C ) ,
UZ)
(11)
When Equation 11 is solved for E t c , Equation 9 can be solved for Reic as follows [setting K ( & J = 4041:
5 0
0
Clearly,fi, = f x / R e t C . Application of this bame theory (Hanks, 1969) to the outer reg:on gives the following set of equations:
01
02
03
04
05 0-
06
07
00
09
10
Figure 2. Theoretical curves showing Re,, and Reocas a function of = rl/r2 as calculated f r o m Equations 12 and 16, respectively
> Re,,
Curves based on assumption that entire flow fleld is laminar. Reoe for all values of u
dK , d € Em
=
0
=
P(l - 2x2)
Reoc =
+ (1 + 3x2)(40c)*-
6464(1
-
efficient in transferring momentum to the bounclary thari the laminar mode. Sirice m was already greater than unity, this increase in PI necessitates an increase in m. We observed t,hat althoagh Equation 8 is written for h = it' is rionetheless valid for any A, being simply a stat'ement of the over-all force balance on the system. Therefore, one can solve Equation 8 in general for A * and obt8ain
x,
u)
(fTe)Fo(to3
When these two sets of equations are solved for a series of values of u and the results plotted as Re, us. U , Figure 2 is obtained. This figure reveals the striking fact t'hat t'he curve of Reoc(u) lies everywhere above t h a t for Rei,(u), clearly showing that according to this theory the inner region of flow is the least, stable of the two when both are in lnniiiiar flow. This observation leads t'o the coiiclusioii that a t a Reynolds numller R e = Reic the iiiner region, upon experiencing a proper disturbance, will undergo traiisition to turbulent flow. However, a t the over-all conditioii of flow a t which chis inner region transition occurs, the outer region is still stable, since Retc < Reoc,as indicated by Figure 2 . Therefore, while the iiiner region undergoes transition, this theory predicts that the outer region remains laminar, thus creating a range of dual flow in which the two niacroscopic zoiies of the flow coexist simultaneously, olie being turbulent' and the other laminar. T h e above conclusion has several significant and predictable consequeiices. Clearly from Equations 15 and 16 to,, and hence Reo,, depends upon A, which in turn is determined by t,he condition ui(E = A) = uo(5 = A). However, since u t ( R e 2 Rete) is no longer given by Equation 2 , it follows t'liat A is no longer given by Equation 4. Similarly, Equation 6 is no longer valid, and consequently, Equation 16 and the upper curve in Figure 2 are 110 longer valid. The significance of this consequence is that X must' shift from its laminar value in order for the stress field P,, to readjust itself to the new mode of momentum transport developed by t,he presence of turbulence in the inner region. Furthermore, we can deduce how it must shift from Equation 8. If t,he inner region traiisists to turbulence at, R e = Rei, as predicted by the t,heory, the wall shear st'ress, Pi, must increase in magnitude because of the change in mechanism of momentum transport to a turbulent mode which is more
U(U 1
1
+ m)
+ am
(17)
From Equation 17 one may readily calculate
Equation 18 clearly shows that A is a nionatoiiic iiicreasiiig function of m. Therefore, when m increases because of the transition of the inner region to turbulence, must incre This conclusion is in direct cont'radiction to the experimeiital results reported in the literature (Crooy and Rothfus, 1962; Rothfus et al., 1966; Walker, 1957), suggesting that not only were their high Reynolds number X data in error (as showii by Brighton and Jones, 1964), but probably their entire range of X values must. be subject to some undetected experimental errors as well. This result leads to still a further consequeiice. If A increases above X a t R e = Retc, then Equation 7 indicates t h a t the shear stress distribution everywhere throughout t'he inner region must increase relative to symmetrically spaced locations in the outer region. Some of the axial momentum which was being transported toward the outer boundary (from the region 5 < E; < A) must now be transported toward the inner bouiidary to compensate for this increase. Consequently, P2 should increase less rapidly with increasing R e thxn it would if the inner region had remained laminar as R e increasedt,hat is, the outer region will be stabilized by the transition of the inner region. The significance of this result is that when the outer region finally does become unstable, the value of Reocwill exceed that given in Figure 2 . Finally, the transition of the inner region must result in a n increase in axial pressure loss due to the increase in the stress distribution there. This increased pressure loss, however, \\-ill not, be so great as vioulcl occur lvit,h transition of t'he full stream to turbulence, because of the stabilization of the laminar outer region which siphons some of its niomeiit'uni off into Ind. Eng. Chem. Fundam., Val. 10, No. 1 , 1971
107
where Rej* = r2ui*p/p, ui* = d P i / p , Reic* is the value of R e t * when R e = Reie, and Bi is a n empirical parameter (Gill and Scher, 1961; Hanks, 1968) which reflects the effect of wall damping on t'urbuleiit eddies. Bi may possibly be a function of u (Wilson and Medn.ell, 1967). I n terms of paramet'er R, one may express Equation 22 as
I
'??Er
~
Reic
Dual Flow
j
I
Turbulent Flow
4i
the inner region. Consequently, if one plots the product of j R e against, Re, a curve similar to the schematic one shown in Figure 3 is to he expected. Two dist'inct breaks in the curve, corresponding to the inner region transition a t Reic and the outer region transition of Reoe, should be observable. Furthermore, the separation between these two breaks both horizontally and vertically should be expected to vary as a function of U , We designate Relc as the first transition or lower critical transition Reynolds number and Reoc as the second transition or upper crit'ical transition Reynolds niimber. SECOSDTRASSITIOK. The occurrence of the inner region transition was seen above to exert' considerable influence on the stability characteristics of the outer region. Because of this, the previous calculation of Reoc is invalid. I n order to compute Reocj one now requires a different expression for ui(€) than that given by Equation 2 , one that. reflects the transitional character of the flow.. This expression in turn, when used with Equation 3 in the condition t i i @ ) = uo(X), will permit calculation of the new values of X. If t'hese new values of X are used in place of 1, Equations 14 and 15 are still valid. However, Equation 13 must' be revised a s follows R'
=
-~ F O ( 4 , X ) 16(1 - u ) ~
u'
(20) where Li is a modified mixing length (Hanks, 1968) which for this geometry may be defined a s
(23)
dz.
where Ri, = Reic I t is convenient to rewrite Equation 20 in dimensionless form as
where E , cxp[-$~~(E- u)]. Since Equation 24 is quadratic in d u J d t , one may solve it as such to obtain (25) gt(E,R)
=
+ [ l + a(€ -
~-
1
(1%u)2(X2
E2)/E
- E2)(1 - Ei)2/E1"2 (26)
with a F k2R2/2(1 25 gives
X formal integration of Equation
G ) ~ ,
which shows the integral's dependence on R as a parameter. The value of X is determined from the continuity of uiand uoa t = X and is the root of the equation
(21) kr2(4 - u ) { l - exp[-+i(t - u ) ] } I n Equation 21 12 = 0.36 and 4 j is a parameter which accounts for the degree of turbulent eddy activity in the flow and may be written for this system in the form
0
=
J'gdE,R)dE
- l/~[l/2 (1 - A')
+ X2 In X I
(28)
The final result which u e need in order to be able to calculate Reoeis an expression for f R e , since Reoc = Roc2/(jRe)oc. If we use Equations 3 and 27 to compute (uz), we can obtain the expression jRe
=
2(1 - u)211 - u ' ) JA(Xz
- E2)g,((,R)d(+ ( 1 6 ) [(l - X2)(1 - 3X2) - 4X4 ln A ] (29)
which is valid for Ric 2 R 5 Roc, as is Equation 28. 'These last two equations also permit one to compute f R e as a function of Re for the range Ric 2 R 5 Roc. The computation of Reocis iiow seen to be straightforward, but tedious. One must simultaneously satisfy Equations 28, 15, and 19 (rearranged as follows) :
=
108 Ind. Eng. Chem. Fundarn., Vol. 10, No. 1, 1971
G)
(19)
where R = R e d i Equation 19 reveals t,he interesting fact that in the outer region one can 110 longer compute Reoc directly, but, rather, must compute Roc. Parameter R is characteristic of turbulent flow analyses (Hanks, 1968 ; Wilson arid ;\ledwell, 1967). Before Roccan be calculated, one must derive an expression from which X may be computed. This in turn requires a n analysis of the transitional flow field in the inlier region. T o do this, we employ an approximate transitional flow theory (Hanks, 1968) which has proved successful in pipes and parallel plat'es, both limiting cases of concentric annuli. I n terms of this theory one may write for the inlier region
L(
Ric dx2-
2Bi(1 - U ) d a G ( l -
Reo,
Re Figure 3. Schematic curve of fRe as a function of Reynolds number, illustrating two critical Reynolds numbers and increased friction losses in dual transitional flow zone
K(E)
=
Roc'
=
6464(1 - u ) ~ Fo ( E o c , ~ o c )
where Xoc is the value of X a t this condition. Transitional and Turbulent Flow. The equations describing the first and second transitions and the dual flow regime between them have been derived above. 811 that
remains for a complete description of the system is to derive a set of equations describing the transitional flow above the second transition Reynolds number. To do this, we again use the approximate transitional flow theory (Hanks. 1968) to describe the stress field, which becomes
0
=
- _t2_ -- X2 E
+
i
i
i
i
exp [ - +,( 1 -
+,=
Roc 2B,(1 -
[
u)
1- A 2(1 -
-
[)2(1 - E , ) 2
e) 1, and
(- $)'
[
1 - ~22]"2 1 - A,
o ~ ) ] " ' { ~
-
,
(31)
1) (32 1
I annulus
(E2 - X?/t X2)(1
-
E)2(1
-#
Specific details elsewhere (Bonner, 1969)
i
,-
20
+ [l + a(t2 -
!
flow Pump instruments Figure 4. Schematic diagram of flow loop used in experimental study
(33)
1
I
a....Q
In Equation 32 B , is a damping parameter (Hanks, 1968), which may be dependent upon u (Wilson and P\ledwell, 1967). As before, Equation 31 is quadratic in (-duo/&) and may be solved as follows:
go(t,R) =
pressure linstrurnents
fRe
2k2(1 - u)ReZ __ (I RZ where E,
0.,
4(1 - u)'
-i
r --
--__
- E0)2/fI1'* (34)
k2R2/2(1 - u ) 3 as before. Cpon formal integrawhere CY tion of Equation 33 one obtains uoas
-
(35) When R
> Roc,the value of X is given by the equation
and f R e is given by fRe
=
1
(A2
2(1 d )_ ( 1 -_u)2 ~ _ ~ _ _ ~._ ~ _ (37)
_
- f9gdtJWt
+
1'
(E2 - X2)g,(t,R)&
The sets of equations debcribing the transitional velocity fields in the two regions of the flow go over smoothly t o the turbulence equations of Wilson and Meciwell (1967) when R >> Roc. Thus, the present' set of equations, anti heuce the model, is complete. Experimental Study
RIost of the existing data for frict'ion factors in the literat'ure for studies which iiicliided the transitional flow region were either so sparse or so badly scattered as to he essentially useless for quantitative purposes, although a few sets appear to be useful for qualitative purposes. hlthough extenai~-evelocity profile data exist (Croop, 1958; Croop aiid Rothfns, 1962; Walker, 1957) and are very interesting, the arguments above conccrning the dependence of X on m cast serious doubt8 on their validity. Therefore, in order to exaniine some of the predictions which follow from the above theory, it \\-as felt necessary to perform both friction loss (Bonner, 1969) and velocity profile measurements. For this purpose, a recirculating flow loop, shown schematically in Figure 4,was used. Detailed specifications of this system are aviiilable (Bonner, 1969).
Flow rates were nieai;urctl iibing a cn1ibr:itetl C I I Xt'iirbiiie flonnieter nioiiitored coiitinuoiislj- b y a Hen-lett-1':ickard electronic counter. Prewire drol~swere measui.ed uiiiig a Pace-JTianco diff erentinl pre5sure tranducer nioiiitored by a coiitiiiuoiis display IIewlett-l'nckard digitxl voltmeter. 'Tu obtain large readiiigs of both iiistrumenti; iii the range of intered, a series of solutions of Polygl-col in watcr, having viscosities which varied :is :I function of Ivater conceiitration from 4 to 5 to 60 to 65 :it, 25OC, \vas used. A series of five ~ cores of vari(iii3 dinnieters annuli was constriicted I J1il:tciiig i n d e the 1-iiich-id. :iliiniiiiuni 11ipethat sen-ed as the outer pipe for all annuli. The cores \yere aligned by 11ti4ng tlieni through perforated disks a t each eitd of the pipe, whic*hh:id holes of diameter c c p d to thf core chnietcr drilled in their centers, tliiis nwuring conce~itricitya t the r i d s . The corcs were put under axial teniion by nieans of a nut and thread arraiigeiiient on one eiid. The position uf the core a t the preesure-inea~iiriiiglocations W:I+ deterniined by depth gage nieasurenients made through the ~irrssuretap holes. K h e n the distance het~veeiithe n-all and the (YJP, as inensured this way, \vas equal to the c~nlculatedanioiuit, thc core 17a> nssiinied to be coiic*eiitrie. Meaeurenient~ were made in low Reynolds iiuniher laminar flow first, to dctrrmiiie n-hrther the system was caapahle of ~)roducingresult:, in agreement n.ith the accepted theoretical laminar values. 13rc:iuse of eqiiilnnciit limitations, turbuleiit floiv nieasurcmeiit~could iiot be made with the sanie fluid used for laminar flo~vmCasiirenients. (81)
-1series of nieaiureniciits u x s made I\-itli no ('ore? in pl:ice aiitl the d:it:i are c~)nqi:iredn.ith the theoieticnl c i ~ r v ~for+ empty i)ii)e,i (Hank>>106s) iii Figure 5 . Tlic>c tl:tla :igree ne11 v.itli thc referrlice e1lrves, iti(1ir:xting t h a t , L i t I t S L r G t for the range covt.rcc1, thc alilipmctlt ~ierforiii~tl >ati-f:ic,torily. Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
109
l
I
i
I
l
l
l
I
I
I
I
I
I
I
I
I
1
l
34
t 32
30
I
O L J I 03 04
I 05
I
I
I
06
07
00
09
o
I
0
F-
c
IO
E Figure 6. Comparison of theoretical velocity profile in laminar flow with experimental data for u = 0.333,Re =
555 0 -
Measured values obtained in present study Computed from Equations 2 and 3
0
1000
2000
3000
4000
Re
Figure 9. Friction factor data for annulus with u = 0.1 074 For experimental data % f = 22.69 f 0.31 for data having Re Theoretical value of % = 22.42 Computed theoretically as described in text
< 1500.
-
301
0 0
28
0 0
0
c 24c
34
I
0
32 r l ; R* +22
0
30
20 1000
0
Figure
2000 Re
3000
4000 2$
7. Friction factor data for annulus with
u =
0.0295
For experimental data f K = 21.34 & 0.18 for data having Re Theoreticalvalue of fRe = 21.01 Computed theoretically as described in text
332 4
< 900. 24-
0
y
1000
2000
4000
3000
Re
Figure 10.
Friction factor data for annulus with u =
0.2238 Far experimental data fRe = 23.1 5 f 0.30 for data having Re Theoretical value of f G = 23.1 9 Computed theoretically as described in text
< 1900.
-
26
: : ,
c
32
--L-L 0
1000
2000
3000
4000
c
Re
Figure 8.
Friction factor data for annulus with u = 0.0455
For experimental data f& = 21.49 iz 0.24 for data having Re Theoretical value of fRe = 21.47. Computed theoretically as described in text
< 1250.
0
2*1
LT W30t
-
n
Reic
1:
26
A series of velocity distributions was measured in a separate annulus using a bent-tube, forward-facing Pitot tube to determine mean velocity profiles. X sample of the laminar velocity data so obtained is shown in Figure 6 in comparison with Equations 2 and 3. Results
Figures 7 through 11 contain the friction loss data obtained with the five different annuli (Bonner, 1969), plotted with the 1 1 0 Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
0
Figure 11.
A
1000
2000 Re
3000
Friction factor data for annulus with u =
0.6699 For experimental data f z = 23.85 iz 0.54 for doto having Re Theoretical value of f% = 23.93 Computed theoretically as described in text
-
< 2100.
I
I
I
I
I
I
I m n 7 m m T l
I
I
0 68-
-
-
; j ; y: -
066-
6
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High Reynolds Number Asymptote
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product as ordinate and R e a s abscissa, as suggested by the theory above. The dual breaks in the data are clearly evident. The statistical mean value of f x e is indicated on each figure, together with a 95y0 confideiice error limit. Also shown is the theoretical value o f f ? for comparison purposes. Figure 12 is a plot of A as ordinate against R e as abscissa for the annulus in which velocity profiles were measured. Figure 13 shows two sets of dat'a obtained in the annulus with u = 0.0455. The t'riangular points correspond to having purposely deflected the outer pipe enough to cause gross ecceiitricit'y of the core a t the pressure-measuring stations. The circular points were obtained with the core as nearly concentric as could be determined by t'he method described above. The solid curves in Figures 7 through 13 correspond to the theory preseiit,ed above, with B t = Bo = 22 chosen to allow agreement with accepted results for highly turbulent flow (Brighton and Jones, 1964; Wilson and Medwell, 1967). These results mere used as the constraint on B e and B o , because no high Reynolds number turbulent dat'a which could be used for this purpose were obtained in the present experiments. Figures 14 and 15 compare experimental and theoretical values of Reic and Reoc,respectively. Discussion of Results
T h e results of the experimental st'udy, t'ogether with the theoretical analysis given earlier, permit us to draw some interest,ing conclusions regarding the nature of the t'raiisition phenomena observed in annuli and the pertinent' variables t h a t influence these phenomena. The data of Figures 7 through 11 clearly show that the behavior predicted in Figure 3, on the basis of the assumption t h a t a two-step transition to turbulence occurs, is observable, suggesting t h a t the proposed mechanism may act'ually occur. If the data of some earlier investigators are plotted in a like manner, bhe same qualitative behavior is suggest'ed, although the statistical scatter of these literature data is often so great as to make quantitative debermination of Redc and Reocdifficult or impossible. The data of Figure 12 clearly reveal the direction in which
4
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a no A n 0
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Figure 13. Two sets of friction factor data for annulus with u = 0.0455 A Measured values with core highly eccentric 0 Measured values with core concentric
- Computed theoretically as described in text
A moves relative to i. This direction is also exrctly as predicted by the theory which is based upon the assumption of transition t,o turbulence near the core first. The lack of quant'itat'ive agreement between the dat'a and the theory is significant'a i d is discussed in detail below. The dat'a obtained in the present study qualitatively agree with the theoretical curves with respect to the shape of their distributions, but fail to agree quantitatively. This lack of quantitative agreement actually reveals some previously unsuspected details about, the nature of the transition phenomena in aiiiiuli. To understand why this is so, one must examine t'he data in Figure 13 rather carefully. Effect of Core Eccentricity on Transition. T h e d a t a presented in Figure 13 correspond to a deliberately achieved eccentricity of t'he core in t8heregion where pressure nienauremerits were made. The triangular points were obt'ained while the outer pipe was displaced sideways by a setscrew mounted externally to the system The displacement' was Ind. Eng. Chem. Fundam., Val. 10, No. 1 , 1971
111
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r Figure 14. Comparison of experimental values of Rezc with theoretical curves as a function of u 0 Measured values b y present authors A Token from data of Walker ( 1 957) Computed from Equation 1 2
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too early, t'hey rise just a lit'tle t'oo high, and they break the second time just a little too soon. This behavior therefore suggests that the method used to determine concentricity is not adequate to remove all eccent'ricity aiid more sophisticated techniques requiring greater care and precision of alignment are required. From the above data, it appears that even slight amounts of core eccentricity, which go unnoticed in laminar and t'urbulent flow, have a large influence on the transition phenomena. This observation accounts for the extreme variability found in literature data which span the transition range. From the above discussion, it appears that t'he deviations from laminar flow observed in the data of Figure 12 starting a t Re = 800 are probably due to core eccent'ricity. Therefore, although these data are qualitatively useful in showing brends, they cannot be quantitatively compared with theoretical curves. Conclusions
a 0
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Figure 15. Comparison of experimental values of Reoe with theoretical curve as a function of 0 Measured volues b y present authors A Taken from data of Walker (1957)
- Computed theoretically as described in text enongh to cause t,he core nearly to come in contact with the wall betwecn the Iressure taps. The circular data points mere obtained with the core as nearly coricerit'ric as possible as measured by t'he technique described. These data reveal several features of significance. First, the laminar flow data for the eccentric case lie a little below the coilcentric data, in accord with other observers' results (Jonssori, 1965). The departure of f R e from f x e occurs very much sooner for the triangular than for the circular points. The first departure for the eccentric case occurs a t about Re = 800, whereas for t'he concentric case it does not occur until Re = 1320. The theoretical value is 1400. K o t only do the triangular data points indicate too early a transition iii the inner region, but they rise to a higher intermediate level of the ordinate than the data for the concentric case. A third feature of the eccent>ric core data is that the second major break point in the f R e curve also occurs much earlier than the corresponding point in the curve for concentric core data. These t,hree characteristic types of behavior are all apparently the result of the pronounced eccentricity of the core. If we now examine the circular data points in relation to the theoretical curve, it appears that all three of the above eccentricity characteristics are evidenced but in a lesser degree-that is, the data break from laminar flow just a little 112 Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
We can conclude that the excursion of the f R e curve from the laminar value in the t,ransitioii region is due to the transition of the inner region to turbulence while the out'er region simultaneously remains in laminar flow. We can also conclude that during this t'ransitjori the value of X first moves from the laminar position 1toward the outer boundary, goes through a maximum excursion a s the outer region experiences transition to turbulence, and then rapidly decreases toward its high Reynolds number asymptote, which lies between X and u . The magnitude of this maximum excursion arid the Reynolds numbers a t which it occurs vary as a function of u . As a result of the transition of the inner region to turbulence, the outer region is stabilized and persists in laminar flow longer than it would if tmheinner region transition had not, occurred. The eccentricity of the aiiiiulus core, while having a minor effect in laminar flow or high Reynolds number turbulent L pronounced effect on the quantitative behavior in the transition region. I n particular, the effect is to decrease both critical Reynolds numbers and increase the excursion of f R e from the laminar value over its theoretical values. The import'ance of this observation is that, from a practical point of view, it is not possible to predict the two critical Reynolds numbers t'heoretically with a very great degree of accuracy because of the 1)roiiouiiced'influence of small core eccentricities. Therefore, although the theory permits us to explain and understand the numerous peculiar results that may be observed in the transition process, one must resort to empirical observations on a particular annulus if he desires a precise value of the two critical Reynolds numbers. I s seen from the data in Figures 14 aiid 15, the experimeiit'al data frequently fall below the theoretical values, no doubt because of minor core eccent'ricitieswhich are all but unavoidable. I n view of this last observation, the empirical calculation method recently proposed by Rothfus and Newby (1970), while correct in principle, is of questionable utility for practical considerations because of the uncertainty of even the best available friction loss data in the transition region. Acknowledgment
The senior author expresses appreciation to the BYU Engineering Analysis Center for making the service available to him while on sabbatical leave. The authors thank Brent W. Sears, who made the velocity profile measurements quoted herein.
literature Cited
Bird, It. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 51, Wiley, Xew York, 1960. Bonner, W.F., 31. S. the&, Brigham Young University, 1969. Brighton, J. .4.) Jones, J. B., J . Raszc Eng., Trans A S M E 86 (4), 835 (1964). Clump, C. W., Kwasnoski, I]., A.I.Ch.E. 3. 14,164 (1968). Croop, E. J., Ph.D. thesis in chemical engineering, Carnegie Institute of Technology, 1958. Croop, E. J.,Rothfus, R. IT., A.I.Ch.E. J . 8,26 (1962). Gil1,W. N., Scher, Marvin, A.I.Ch.E.J. 7,61 (1961). Hanks, R.WT’.,.4.2.Ch.E.J. 9,45 (1963). Hanks, 1LN7.,ii.I.Ch.E.J. 14,691 (1968). Hanks, R. W., A.I.Ch.E.3. 15,2.5 (1969). Jonsson. V. K.. Ph.D. thesis. University of 3Iinnesota. 1965. Lamb, H., “Hydrodynarnids,” 6th ed”., p. 586, Cambridge University Press, London, 1932. Lea, F. C., Tadros, A. G., Phzl. Illag. 11 (Ser. 7), 1235 (1931). Lorenz, F. IT., “Uber turbulente Stromung durch Rohre mit kreisringformigen Querschnitt,” Mitt. Inst. Stromungsmachinen, Tech. Hochschule Karlsruhe, Heft 2, pp. 26-66, 1932.
hlacagno, E. 0.)McDougall, D. W., A.I.Ch.E.J. 12,437 (1966). Itatkowsky, D . A , , Can. 3 . Chem. Eng., 44,s (1966). Rothfuq, 11. R., Ph.D. thesis in chemical engineering, Carnegie Institute of Technologv. 1948. Rothfus, 11. It., 3londridj C. C., Senecal, V. E., 1nd. Eng. Cheni. 42,2311 (1950). Rothfus, 11. R.,Newby, R.A.,A.l.Ch.E.J. 16,173 (1970). Rothfus. R. R.. Sartorv, W.K.. Kermode, It. I., A.I.Ch.E.J. 12. 1086 (1966). Rothfus. R. li.. Walker. J. E.. Whan. G. A , . d . I . C h . E . J . 4. Walker, J. E., Ph.D. thesis in chemical engineering, Carnegie Institute of Technology, 1957. Wilson, K IT., filedwell, J. 0.) ASIIE Paper 67-WA/HT-4, ASAIE Pittsburgh hleeting, P i t t h r g h , Pa , Kov. 12-17, 1967. RI.:CEIVED for review !day ~ C E P T I : D November
6, 1970 9, 1970
Work supported by NSF grants GK-1922 and GK-15893.
Macromolecular Ultrafiltration with Microporous Membranes Robert 1. Goldsmith Abcor, Inc., Cambridge, Mass. 02139
The mechanism of macromolecular ultrafiltration with microporous membranes i s discussed, focusing on factors that control membrane flux and solute retention. Flux i s limited b y mass transfer conditions on the feed solution side of the membrane (concentration polarization). Solute retention i s determined by geometric properties of the membrane pores and the macromolecules in solution, as well as concentration polarization. Ultrafiltration data for solutions of Dextran fractions and a Carbowax fraction are presented and compared with theory. Agreement for turbulent flow, laminar flow, and stirred cell ultrafiltration systems i s good.
uring the past decade ult’rafiltratiori has been advanced
D from a laboratory curiosity to a n important industrial
unit operation. Practical applications include a broad spectrum of solution concentrations and/or fractionations. I n an ultrafiltration process (Figure 1) a feed solution is introduced into a membrane unit, where solvent and certain solutes pass through the membrane under a n applied hydrost’atic pressure. Solutes unable to pass the membrane are retained, concentrated, and removed in solution. The porosity of the membrane determines, primarily on the basis of size, which solutes pass through and which are retained b y the membrane. The pore structure of this molecular sieve is such that i t is inherently “nonplugging,” and stable fluxes for long-term operation are achievable. RIany ultrafiltration applications involve the retent,ion of relatively high molecular weight solutes, accompanied by the removal t’hrough the membrane of lower molecular weight impurities. Of current interest are the coiiceiitratioii and purification of enzyme solutions, and the fracbioiiation of cheese whey for protein recovery (deFilippi and Goldsmith, 1970). For this t’ype of operation, the use of a high-flux membrane leads to low-pressure operation, frequently below 50 psi. As 110 phase change occurs during ultrafiltration, i t offers several attractions. Energy requirements for concentration by
ult,rafiltration, compared with those for evaporation, are relatively loiv. In addition, sensitive macrosolutes such as funct,ional proteills may be treated without denaturatiori. This paper examines ultrafilt,ration parameters which determine ultrafiltration rate and select’ivity.Data for turbulent flow, laminar flow, and stirred-cell membrane systems are presented and compared with theoretical predictions. Experimental
Ultrafiltration experiments were 1)erformed with membranes in three different ultrafiltration systems. A flow schematic, identical for each system, is shown in Figure 2. Solutions from a feed reservoir were pumped through the ultrafiltration system. Inlet and/or outlet pressures were measured with pressure gages to within 0.5 psi, and the retentate flow was measured with a rotameter. Ultrafiltration rates were measured volumetrically. The operating pressure was controlled with a back-pressure regulator. 130th retentate and permeate solutions were recycled t o the feed reservoir. The feed solution flow was substantially greater than the ultrafiltration rate, so that differential operation was achieved -Le., feed and retentate solution flows and concentrations were essentially equal. The feed reservoir temperature was maintained within 1°C by immersion of the reservoir in a coiistant temperature bath. Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
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