Mixing effects in vertical upflow bubble columns

The Chemical Engzneermg Journal, 25 (1982)

29 - 38

29

Mixing Effects in Vertical Upflow Bubble Columns MARIE &HOVA Znstztute of Hydrodynamccs,

Czechoslovak Academy

of Sciences, Prague 6 - Podbaba (Czechoslouakza)

(Received 11 February 1981, m final form 9 November 1981)

Abstract The phenomenon of a rrsmg macrovortex of preferentially aerated llquld was observed m large-scale bubble columns A theoretuzal explanation of this phenomenon 1s presented, based on the theory of the turbulent free boundary layer The descrlptlon results m a single-parameter model, based on the coefficient of Prandtl’s mlxmg length from the phenomenologlcal theory of turbulence after mtroducmg acceptable slmpllflcatlons The expenmental procedure of measurmg the coefficients of Prandtl’s mlxmg length from the boundaries of Jets forming after mtroductlon of a tracer fluid mto a macrovortex and into single-phase flow has been devised Finally, a mixing coefficient 1s defined, relating the degree of mlxmg m the macrovortex to the turbulence created m smglephase flow alone. Thus a dlmenslonless smgleparameter model of non-ideal flow m a part of the total flow in a bubble column, due essentially to gas maldtstrlbutlon, has been formula ted

1 THE THEORY

AND ITS APPLICATION

1 1 Introduction

Bubble columns of any appreciable pilotplant or commercial size represent a complicated hydrodynamic system, difficult, if at all possible, to describe by methods common to other types of equipment [ 1 - 41. We made an extensive series of observations on these systems with their possible apphcation to the fermentation industry, mamly for biomass production [ 51. A very good understanding of the phenomena takmg place m tower fermentors is necessary m their largescale application. A certam phenomenon, presented m a simphfied and slightly idealized 0300-9467/82/0000-0000/$02

75

Fig 1 Idealized view of a macrovortex forming m large-scale bubble columns I, macrovortex region, II, backmixed sections, III, deadwater section, 1, gas sparger, 2, nozzle orifice, 3, free Jet region, 4, macrovortex sectlon mvestlgated, 5, tracer dosing device

view m Fig. 1, proved to occur constantly m all columns havmg diameters larger than 0.2 m. It has been termed the nsmg macrovortex, which propagates from the gas distnbutor as far as the liquid level m the column. Going more deeply mto the problem, three distinct regions can be defined m alI larger scale bubble columns, due mamly to gas maldistnbution m large-size spargers (I) The aforementioned curved macrovortex of fluid richer m gas than the average gas holdup, movmg upward at an mstantaneous velocity larger than the average, yet essentially constant This is a case of forward mixing. 0 Elsevler Sequola/Prmted

m The Netherlands

30

(II) Backflow of some parts of the hquld havmg gas content smaller than the average gas holdup, agamst the average superflclal velocity. In some cases reclrculatlon has been observed, leadmg to (III) deadwater zones or nearly stagnant re@ons. The present contnbutlon 1sdevoted to an attempt at a rather more profound explanatlon of the phenomenon of the nsmg macrovortex using standard methods from the theory of turbulence m a special case. We assume that an analogy exists with the submerged Jet m angle-phase flow, which 1s regarded as the log~al hmltmg case of the two-phase system. When a Jet 1s introduced mto the single-phase system, It expands and the Jet boundary is linear by defmltlon. From the size of the submerged Jet, basic turbulence charactenstlcs can be defined. Now, m the two-phase system The Jet boundanes m the macrovortex follow its streamlines, and are therefore always curvllmear. If we define a suitable, more or less generally vahd coordinate system, m which the Jet boundanes are linear, we can draw an analogy between both these cases and use the identified turbulence charactenstlcs m the case observed and examined. Based on these conaderatlons, the followmg procedure of mvestlgatmg the problem was adopted (a) a suitable curvilinear coordinate system, capable of hneanzmg the observed boundanes of a submerged Jet introduced into the nsmg macrovortex stream, must be sought; (b) a suitable parameter from the phenomenoloscal theory of turbulence must be defined and used to apply the theoretical background denved to the defmltlon of mlxmg behavlour m single- and two-phase flows, (c) a method of obtammg the parameter from expenmental data must be devised to make the model work, (d) and, eventually an attempt can be made at correlating the parameter mth known hydrodynamic magmtudes. 1 2 Theoretical of turbulence

solution

based on the theory

Returning to Fig. 1, it is advantageous to introduce into the system exammed a threedimensional Euchdean space E3 mth the

Carteaan coordmate system E(X1,x2, x3) and to make the equipment axis comclde with the dlrectlon of the axis 3~~.Wlthm this system, valid for the entire vessel, we define a subsystem, the macrovortex (I). In order to mvestlgate this subsystem both theoretlcally and visually, we introduce tracer fluid from a submerged nozzle which forms a Jet expandmg m the macrovortex, designed as the space !ZJ This space 1scharacterized by the mean arc C. When using its parametnc descnptlon fl(ol)

Xl

=

x2

= f2(@

x3

= f3(4

(1) (-

0)

where 0 denotes the length of the arc C, and

(2) we can define the relation between the two coordmate systems [ 51

gl(%P,Y) = fl(N

--P

!?2(&

+ P $;

P, 7)

= f2(@

g3@,P,Y) = P sinY

;!$

(a)

cm

Y

(=

Xl)

(4

cm

Y

(=

x2)

(3)

(=x3)

m the mterval y3 = (al,a3) X (0, b,) X (0,27r). For a special choice of the real number b,, eqns. (3) determme a curvlhnear orthogonal coordinate system e(a,P,r) on the open set ii: (C E3) with the coefficients of hneanzatlon h,=l--pcosyR(cu) h2 = 1

(4)

&=P fig 1s then an unage of a regular and one-toone mapping -C3of the interval I, and the boundary K(GJ) is the ‘channel area’ having the mean arc C and bounded by the normal areas of the curve at the points [a,, 0, 0] and [a3,0,01.

The ongm of the arc C 1s situated at the centre of the nozzle opening, from which the tracer ISintroduced mto the channel area in the macrovortex. The part of the latter, thus vlsuahzed, 1smade avtiable to observation as an integral part of the total system exammed,

31

to which the coordinate system ~(a, j3,y) may be apphed. It is not essential to apply further treatment to this generally defined curvllmear system, yet it also proved to be possible, by suitable application of the theory of analytical functions [6], to fmd an analytical descnption of the curvature C. This facilitates the computational procedure m special cases and is reported elsewhere [ 71. Details on the denvatlons necessary to define the curvilinear coordmate system can be found m ref. 5. In our further treatment presented here, we shall restnct ourselves to the case of a free Jet S2s, axlsymmetncal with respect to C. It has readily been proved [ 51 that there is a distmct analogy between the curvrhnear coordmate system and the cyhndncal one, when the linear longitudmal coordmate 3c1is replaced by its curvilmear analogy o. At this point, we must mtroduce some restnctmg conditions, to make further treatment possible As will be seen, the conditions introduced are not more restnctive than those generally accepted, even m advanced hydrodynamical considerations (1) Fluid flows m the steady state and has constant density pL, viscosity pL and superficial velocity uL m the direction of the axlS

a.

(2) Fluid is homogeneous m the space St:, there is no influence of external forces and it flows under constant hydrostatic pressure p. (3) The liquid source is introduced into the mam stream of the macrovortex and has close physical properties, viz, pS 2 pL, p, = pL, . . , index s referrmg to the properties of the tracer hquid, it flows m the direction of the axis X, and does not adhere to the nozzle wall. (4) The source introduced generates a free Jet (an,) of mcompressible viscous fluid emerging from a circular onfice with diameter r, (S /~,,(a~))mto the essentially homogeneous fluid surroundmgs of the mam stream. (5) r, B rL 9 r, (rL = b,) must be valid. (6) The stream from the physical nozzle forms the zone of amsotropic mhomogeneous free turbulence, having properties close to the phenomena described by the theory of the turbulent free boundary layer m turbulent flow. (7) There are no transient phenomena affecting the observations after tracer mJection

Now, bearmg ail these restnctlons m mind, we can write.

(6) and the

ff a,0 E 12 (’ (a,,~,) X (0,/i,), boundary conditions &(a,8

= G, max(o)

q3(%PI = 0

QAc%P) = GL(4

(’

)

v a E (a*, a*>

I

P=O v’a E

UL)

t

&WV = 0 &(%P) = w3

(’ 4)

P = hot@

(W

o! = a1

i

&3(a) = 0

(Ul,UJ

P E (0, how

For reasons of adequate mathematical formulation, we have to add the followmg condition.

&(%P) = U,(P) Q(%P) =

U,(P)

I

a =

u2,

/3 E to,

ho(u2)> (7b)

where ii cY,max(~), 7:.p= 2. p@,PAhow, U,(P), and U,(p) are known functions, uL, U, (uL, u, > 0) are given constants and (a r, u2>is the interval exammed. The function sought, ii,, should, apart from the necessary conditions, also fulfll aii, -

= 0

ap

for

o E (u1,u2)

(7c) p = 0 (or 0 = ho(a)) The Leibenzen integral condition, correspondmg to the system defined by eqns. (5) and (6), is as follows a

--

ho

s

U,(u,

- ii,)/3 dfl = 0

V o E (ur,u2>

(3) This system of partial differential equations descnbmg the free turbulent boundary layer m a homogeneous stream is farrly general, and can only be solved after mtroducmg the conditions for particular cases. 0

1 3 Appllcutlon of the theory macrovortex region

to the rwng

Several problems must be solved dunng the phenomenological treatment of this problem :

32

(a) The boundary of the submerged Jet 1s not smooth, due to local eddies forming at it, for further treatment, smoothed values must be used (b) Generally, the basic section of the Jet, 1 e. the regon of developed turbulence, must be determined when applymg expenmental condltlons to eqns. (5) - (8) In the following conslderatlons we assume that the system of relations is valid m the entire re@on of the free Jet. This means that the potential core and transltlon zones are neglected [8]. When attempting to descnbe the free Jet according to eqns. (5) - (8), we do not know one of the boundary condltlons, ii,(cu,p) = iiol,max(~) m the Jet axis, nor the Reynolds stress and the functions U1 and U,. Exact analytical sol&on 1s therefore not possible We have to resort to the procedure common in phenomenolo@cal theones of turbulence and introduce an expenmentally determined element into the system consldered. Knowing the lme boundmg the free Jet from As observation, the concept of the Prandtl murmg length, as applied m refs. 9 and 10 to the theory of momentum transfer, can be used. Let us introduce ii,=&--u~

(9)

and the problem then reduces to the theoretical case of a Jet m a stationary medium

$ WI) + &,) =

0

(5 + =-_

$.i_(?)+ _$p) 2

;v-p~ i 1

@a)

‘p’ (x,/3E I*, under the followmg condltlons

h(%P) = El

(YEhzl,U2),

(7aa)

qd%P) = 0 iil(%P) = 0 = q&G)

p=o

o!

E (Ul,%), P = &(a)

(Tab) Ul(%P) = &.s(P)

a = ai,

pE

(O,i,(U~)) WC)

where

Let us further consider the affinity of velocity profiles. This assumption reduces the partial problem to a system of ordinary dlfferentlal equations. Even this system 1snot solvable for uL f 0, should condltlon (8a) apply simultaneously. The problem is generally solved for two hmltmg cases ii&L%

l-u,SuL

(11)

ii&AL< l-uszuL

(12)

This means that d a stream having the velocity u, enters the surroundmgs which have a constant velocity uL m the dlrectlon of (Y,and us% UL, we have a problem analogous to the first case (eqn (11)) when the thickness of the boundary layer &,(a) mcreases lmearly with the distance from the nozzle [8] . h&v)

(13)

= c(a + a)

where a 1sthe pole distance from the nozzle opening Jet velocity dissipates along the axis according to the hyperbolic rule 2rs ~l.max =B-u CX+u S’

B = const.

(14)

With increasing distance from the nozzle m the duectlon of flow, the velocity ii, decreases, as uL = constant. There exists, therefore, an 6 for which iil(ii,@) < uL. This problem then resembles the second hmltmg case (eqn. (12)), which is, however, not Important from the vlewpomt of apphcatlon. It 1s obvious that all practical mjectlon velocltles u, he wlthm the interval conadered. Prandtl’s conslderatlons on the apphcatlon of the murmg length to the descnptlon of momentum transfer m the theory of the free boundary layer set out from the assumption of lmear increase of Lp with the distance from the nozzle openmg Lp =

~,(a++),

c,

=

const

(15)

and

(16) The proportlonahty constant c, characterizes the ‘degree of turbulence’ for the entire Jet and can be ascertamed only from expenment.

33

The ordmary dlfferentml equation, obtamed from eqns (5a) and (6a) on mtroducmg the affmlte profiles, has been solved by approxlmatlons by Tollmlen [9, lo], and the solution ii Jii 1,max(~,/3) has been expressed by a series m the vlcmlty of the pomts 0 = 0 and p = &,(a) for any arbltranly determined a Using this solution, we get a simple relation h,(a) = 3.4(01+ a)c$/3

(17)

which can be used easily for determmmg c, from eqn (13), should &(a) be known from expenmental data. As has been shown, a phenomenolo@cal element must be introduced mto the theory, nsmg from experimental observations. To make the slmphfled model work for the particular case of a macrovortex nsmg in a bubble column, we have to introduce the followmg condltlons (1) System Cartesian coordmates E(x~,x~,x~) the system ongm 1s situated at the centre of the nozzle openmg, axis 3c1 comcldes with the dlrectlon of the column axlS. (2) CuM1lnea.r coordmate system E(CY, p, y) the ongm of the mean arc C E i23 comcldes with the ongm of the Cartesmn system (al = 0), the parametnzatlon of arc C is @ven by

fi(t) = A*(-$t2 f2(t) = -A*(t

+ it-2

+ 2 In Itl)

(18)

+ 1)2(2/t)

where t E (--1,O) and A* = A2/2u is the parameter, relations (3) which define the system E(CX, 0, y) and the hneanzmg coefflclents (eqns. (4)), and the curvature R(a) result from eqns (18) as follows f;(t) = f2w

fl(4

= fit4

a E (0, a2)

(19)

where a2 1sthe length of the arc C (3) In all relations concerning at3, the velocity uL 1ssubstituted here by

u rel =

%

-

%

+--

UL

Urel =

1 -Eg’

const.

(20)

and the direction of u,,~ 1s the same as the direction of the coordinate CL (4) InJection velocity u, = constant and its dlrectlon 1s the direction of the coordmate Xl.

(5) In the two-phase systems investigated, all data measured must be corrected for the gas holdup, to make eqns. (5) - (8) valid for this system without undue error. Now, to account for the hmltmg case of single-phase flow, the restnctlons pven above simplify to the followmg condltlons (1’) The Cartesian coordinate system is ldentlcal to that of the two-phase system (2’) The curvilinear system e(cr,&r) reduces to the cyhndncal system when (x = x1, y being the polar angle. The arc parametnzatlon 1sthen executed by the simple relations fl(4

= (4

fi(a)

= 0

and

R(cY)+ 0

(3’) The choice of the channel area a$ bl = r,, a3 1s the linear distance from the nozzle openmg to the liquid level, St; 1sthe entlre volume of the column above the nozzle opening. Fluid velocity uL = constant, mJectlon velocity u, = constant, and both have the dlrectlon of the coordinate x1. (5’) Spreading of Jet boundanes obeys the general theory of the free boundary layer without any corrections to eqns. (5) - (8) being needed. 1 4 Defznztlon of the ‘mlxmg coeffzaent CM There 1sa definite disadvantage m practical apphcatlon of the dispersion coefficient to axially dispersed turbulent flow owing to its dunenslonahty. The mtroductlon of some dunenslonless group contammg it is also rather problematic, as the adequate defmltlon of an appropriate length parameter 1s not always easy or Justified. Now, the Prandtl murmg length 1s also dimensional and, m addition, its numerical value depends on the distance from the source mtroduced mto the system, as follows from the defining eqn. (15). This disadvantage can be overcome by defmmg a dnnenslonless magmtude, which can easily be obtamed from expenmental data, and which adequately describes the murmg situation m the macrovortex, VIZ,the ‘muring coefficient’

34

(21)

where (Y is the distance from the nozzle opening. The advantage of this coefficient IS that it relates the random situation created by the mtroduction of gas mto a nsmg liquid stream m part of the macrovortex to the situation existing m smgle-phase flow. The mlxmg coefficient actually defines the murmg mtensity in this stream.

2 EXPERIMENTAL DETAILS, DATA TREATMENT AND DISCUSSION OF RESULTS

2 1 Introduction In the preceding part we applied seemingly rather abstract considerations from the theones of turbulence and the turbulent free boundary layer to a phenomenon observed m large-scale bubble columns. From very general relations, still on a ngorous level, we eventually obtained workmg relations for a coordmate system which allow us to linearize the streamlmes m two-phase systems m which the curved rismg macrovortex occurs. Pernnssible simphfication had to be introduced. A further phenomenological step was necessary to get relations which modelled the situation m bubble columns and the conditions for a particular system have been identified. Eventually, a dnnensionless mlxmg coefficient has been suggested, defmmg the degree of mlxmg m the nsmg macrovortex as bemg due solely to the presence of the gaseous phase. In this section it remams to define workmg expenmental conditions to obtain the parameters which must enter the model from expenmental data, to outline a way of solvmg the model with such data and to give a method for calculatmg the murmg coefficients. 2.2 Expertmen tal The expenmental plant used is sketched m Fig. 2. The buble column was assembled from standard glass tubmg, 250 mm m diameter, and the distance between the dlstnbutor and the liquid level was 3 m. Exchangeable distnbutors were used, mamly perforated plates (openmg dnu-neters 1 - 5 mm) and fntted glass distnbutors with equivalent opening diameters of 0.1 and 0.15 mm Free

1.p

r.t.r

Fig 2 Diagram of the experlmental apparatus and mstrumentatlon CMD, constant-mass dosmg device, GFC, gas flow rate control, GFI, gas flow rate mdlcatlon, LFC, hquld flow rate control, LFI, hquld flow rate mdlcatlon, PPD, perforated plate dlstrlbutor, Cl, C2, cameras with synchronized exposure

sections of the distributors were m the range 0.61 - 3.0%. As the large-scale equipment was used only for venfymg the model developed in Section 1, only water/au systems were used. The liquid level mdication at the top of the column was used to determine the holdup of gas from the difference m levels under gassing and non-aeration. Flow rates of both media were controlled by manually operated control valves GFC and LFC and were mdicated by flowmeters GFI and LFI. Temperature was essentially constant dunng the expenments and equal to that of tap water introduced without recirculation. Pressure was also constant and equal to the static pressure of the aerated or non-aerated liquid column. Tracer was mtroduced steadily at amounts up to 10 ml per mJection by a constant-mass dosing device (CMD). It consisted of a firmly mounted mjection synnge. A constant mass acted on the piston causing its steady downward movement. A rather complicated arrangement was necessary to obtam reliable data on the contours of the submerged let forming after mJection of the tracer mto the liquid bulk. To avoid perspective distortion, two cameras were used with synchronized exposure at nght angles. When the Jet appeared to have

35

boundary after prolectmg the slide at a specified enlargement ratio. Thus, coordmates were determined for points at distances 2.5, 5, 7.5,10,12 5,15,17.5 and 20 cm from the nozzle openmg. Altogether five gaseous velocities (0.717, 1.021, 1.324, 1.642 and 1.965 cm s-l) and five liquid velocities (0.22, 0.373, 0.51, 0.662 and 0.815 cm SK’) were examined for each experimental run, arranged as Graeco-Latin squares. Results of a typical run for the fritted glass distributor having equivalent pore diameter 0.15 mm and a free section of 0.61% are given m Table 1. Analogous tables have been obtamed for all distributor/free-section combmations tested.

(b) Fig 3 Typical Jet formmg m the macrovortex near-hnear boundaries on one of the screens, the exposure was taken After removmg the pictures when this was not nearly absolutely true, the remammg set of let pictures was passed for evaluation. The nozzle could move along the diameter and tip sideways, so that various parts of the column could be traced. Water soluble dyes were used m amounts up to 2 mg per charge, which did not sigmficantly affect the actual system environment. This was sporadically checked by determmmg the bubble size distributions within the Jet and m the hquid bulk m the macrovortex. Photographs of a typical pan are reproduced m black and white m Fig. 3. It proved relatively easy to read off points on the Jet

2 3 Data treatment Data from the experiment supplied mformation on the Jet boundaries m Cartesian coordmates xi, x2 both for two-phase and for single-phase systems. Simultaneously, flow rates uL and u, and gas holdup data, eg, were also obtamed These data were next used to obtam linear contours m the special curvilinear coordinate system E(cv,~) As mentioned before, we cannot devise a feasible method of obtammg the shape of the let axis directly, so an mduect method had to be devised using the Jet contours obtamed straight from the experiments. There are several procedures possible, usmg standard software, the one adopted by us is as follows (1) The let boundary is described analytically [5] in e(xl,x2). (2) All considerations are based on a sectional view, as we take the Jet as axlsymmetrical, therefore only coordmates x 1 and x2 or o and 0 are mentioned. (3) The coordmates of the pomts on the Jet axis (arc C) are determmed m c(x1,x2) as mtersections of arcs parallel to the let boundary. (4) The analytical description of the Jet axis is then computed m e(x1,x2), i.e., by a suitable method the parameter A2/2u from eqn. (18) is determmed. (5) Next the length of the arc C, i.e. a,, 1s computed. (6) The coordmates of the points on the let boundary are obtamed afterwards m the curvllmear coordinate system E(CX, 0) (7) Usmg linear regression, the pomts on the let boundary are smoothed m e(cu,p)

36 TABLE1 ug (cm

s-l)

(cm

s-l)

UL

0

% (ChF CM (c)IF

1021

1324

1642

1965

0017 0184 1000

0020 0204 1169

0220 0025 0 238 1479

0 029 0 259 1 679

0 035 0 268 1766

0 018 0193 1133

0021 0222 1400

0373 0028 0252 1695

0031 0269 1873

0039 0 286 2050

0019 0205 1377

0023 0232 1665

0510 0032 0 265 2026

0040 0284 2 244

0 050 0 301 2453

0021 0232 1905

0026 0252 2152

0662 0032 0278 2 504

0041 0288 2 631

0050 0 305 2868

0 021 0223 2463

0027 0243 2 797

0815 0032 0250 2924

0040 0264 3175

0050 0 283 3 511

0184

us (ems-') % (C)ZF cM

0178

(ChF

UL (ems-') fg (C)ZF cM 0

(ChF

166

us (ems-') % z 0

(ChF z4~

0 717

151

(cm se')

% (ChF CM

0123

fChF

and the averaged tangent of the straight lme resultmg m this coordmate system is obtamed. (8) The coefficient of the Prandtl murmg length is obtained as the function (c,)*F = F(UL,

ad,

= COnA

Actually, this coefficient might Just as well be sufficient to describe the degree of turbulence m the let, it differs, however, for different values of U, To free it from this Influence, it is advantageous to resort to the hmitmg case of smgle-phase flow. For this we obtam, m a straightforward way, the value of at an identical U, In this case, points (%dlF on the Jet boundary he on an essentially straight line m the Cartesian system E(x~,x2). Any linear regression program then renders the tangent of the smoothed boundmg hne and thus the required value of (c,)iF. By dividing (C,.,& by (c,)ir we WentUdly get the mtensity of murmg m the two-phase macrovortex -the mixmg coefficient, which by defmition must always be larger than unity.

2 4 Dwcusslon of results In this contribution we have tned to descnbe adequately, using only absolutely necessary simphfications, the phenomenon observed m large-scale bubble columns, i.e. the freely nsing macrovortex of preferentially aerated hquid Let us start the discussion with some remarks on the validity of the theoretical approach. It was possible to define a curvilinear coordinate system, quite general and capable of hneanzmg the Jet boundaries of a submerged Jet formmg m the exammed system. Moreover, the boundmg line and/or the centrehne of the Jet could be described analytically. Thus, a direct comparison of the turbulence charactenstics m single- and twophase flows became possible. The chosen parameter for the comparison, the Prandtl murmg length, required very little further simphfication, and was found to be suitable and expenmentally available from relatively easily obtamable data. There was, however, an unexpected phenomenon which

37

turned up dunng evaluation of the expenmerits. Should the theory of the free turbulent boundary layer stnctly apply, then after mtroducmg the additional turbuhzmg agent (gas bubbles), and after correcting the system boundanes for gas holdup, a contructzon, rather then expansion, of the Jet boundanes should actually be observed As the expenmental results show, this 1snot the case, and the Jets enlarge after aeration has commenced. To check on the validity of the theory m smgle-phase systems, expenments were carned out on Jet behavlour while the ratio m = uL/u, was increased. Single-phase Jets reacted perfectly m accordance with the theory. In all the expenments carned out both m two-phase and m single-phase systems, only the single value of U, = constant = 87.2 cm s-’ was used consistently. The theory applies then stnctly only to single-phase systems, and the Prandtl mlxmg length obtamed represents the increase m murmg due to the mtroductlon of the tracer stream. On the other hand, for two-phase flows some additional influence must be found to account for the phenomenon observed. A plausible explanation has been found m the analogy to the mlxmg of hqulds by submerged Jets [13] when the entramment plays an important role. Assuming that the streams of bubbles cause mixing m the liquid stream, then the Jets actually must expand as, even theoretically, a larger amount of energy 1s being dissipated m the same volume. Accepting this explanation, the theoretical concept can also be applied to the aerated systems, and the vahdlty of the observed fact that CM > 1 can be Justified. A practical method of obtammg the expenmental parameters necessary for the apphcatlon of the model has also been found It has one dlsadvantage - m its easiest apphcation it 1sessentially restncted to translucent systems or system parts. Theoretically, other sources of signals apart from visible light might help m overcommg this shortcoming.

dependmg on the general hydrodynamic parameters, which can be deduced from the vanables tested, is as follows CM = 0.0493 + 0.7323 X 10m3Re, + + 0.3743 X 10L6 Re,’ + - 0.0707 X 10e6 Rei + + 0.0413 X 1O-6 Re,ReL ‘ct Re, E (550,2038) Re, E (1128,3091) The correlation coefficient is 0.9947 and the small value of the constant indicates that the influence of the Reynolds numbers, actually representing here the fluid velocltles, 1s substantial. Thus the murmg coefficient can be regarded as a reasonably sensitive indicator of changes m flow of the phases m a bubble column. Thus, the evaluation procedure seems to be Justified, as a reasonably accurate correlation results, despite the fact that some smoothmg of the data on Jet boundaries was inevitable. In conclusion, it can be stated that a practical descnptlon of the murmg phenomena occurring m the freely nsmg macrovortex forming m large-scale bubble columns has been obtamed. It has a dlstmct relation to mixing mtenslty, and as such can be used m heat and mass transfer and kinetic conslderatlons m bubble columns. The method of tracer inJection m order to follow the course of the macrovortex and to define the parameters of non-ideal plug flow can easily be extended to fmd the size of the macrovortex and thus the portion of the entire volume occupied by it.

NOMENCLATURE

a, a

b,

bl 2 5 Data correlatzon Just to check how well the data obtamed can be fitted by some form of correlation, data on CM were subJected to non-linear regression for each separate dlstnbutor tested For the data presented here, the relation

and

CM cln

real numbers pole distance from nozzle opening real numbers radius of the channel area mixing coefficient coefficient of Prandtl mlxmg length tangent of Jet boundary diameter coefficients of lineanzation

38 h,(a)

=

LP L

m P R(a) 8, t u

h,

thickness of boundary layer, real function of a real vanable Prandtl mlxmg length regular and one-to-one mappmg = uL/uS, velocity ratio pressure curvature of curve at a point radius Reynolds number parameter in parametnc expression of a curve superficial velocity

Greek symbols eddy vlscoslty gas holdup boundary of set 52, dynamic vlscoslty density turbulent density of momentum flux open set, simple continuous region Indices Ii! z max rel S Z

1F 2F

%

index denommatlon hquld maxunal value relative source equipment angle-phase system two-phase system

index denommatlon Jet boundanes

Other symbols closure of a set, mean value of a function REFERENCES

7 8

9

!iw

0, Y

o,fl

10 11

12

13

0 Levensplel, Chemrcal Reactron Engmeenng, Wiley, New York, 1962 J C Mecklenburgh and S Hartland, The Theory of Backmzxrng, Wley, New York, 1975 Z ,%irb~Eek and M &ichov& m Z &rb&Eek (ed ), Mwrobral Engmeermg, Butterworth, London, 1974, pp 365 - 374 M &ichovl and Z St&b&ek, Chem Eng J, 6 (1973) 195 M &ichovl, Ph D Theses, Prague Institute of Chemical Technology, 1978, 115 pp I Cernjr, Fundamentals of Mathematrcal Analysis rn the Complex Domam, Academia, Prague, 1967 (m Czech) M SQchovl, Acta Tech (Prague), m press G N Abramovlch, Theory of Turbulent Jets, State Pub1 of Phys -Mathem Literature, Moscow, 1960 (m Russlan) W TollmIen, Z Angew Math Mech, 6 (1926) 468 W TollmIen, Handb Exp Phys, Vol IV, Pt 1, Lelpzlg, 1931, p 241 J 0 Hmze, Turbulence -An Introduction to its Mechanrsm and Theory, McGraw-H& New York, 1959 B Hostmskjr, Differentlo Geometry of Curves and Planes, Natural Science Pub1 , Prague, 1950 (m Czech) Z &e’rb@ek and P Tausk, Mxcmg m the Chemlcal Industry, Pergamon Press, Oxford, 1965