A note of the plunging liquid jet reactor

Shorter Communications REFERENCES ARMISTEAD R. A., Jr. and KEYES J. J., Jr.,J. Heat Transfer, 1968 13. BAKEWELL H. P., Jr. and LUMLEY J. L., Physics Fluids, 1967 10, 1880. CARSLAW H. S. and JAEGER J. C., Condition of Heat in Solids (1st Edn.), pp. 40-43, Oxford University Press, 1947. CORINO E. R. and BRODKEY R. S., A visual investigation of the wall region in turbulent flow, West Virginia University- Kanawha Valley Graduate Center, Professional Development Lectures, Vol. 1, No. 3, 1969. [51 DANCKWERTS P. V., Ind. Engng Chem. 195143 1460. F51 HANRATTY T. J.,A.I.Ch.E. JI 19562 359. [71 HARRIOT P., Chem. Enyng Sci. 1962 17 149. 181 HIGBIE R., Trans.Am. Inst. Chem. Engrs 1935 3165. [91 KOPPEL L. B., PATEL R. D. and HOLMES J. T., A.1.Ch.E. Ji 1966 12,941. [lOI KOPPEL L. B., PATEL R. D. and HOLMES J. T., A.1.Ch.E. Jll966 12,947. [111 KOVASY K., Chem. Engng Sci. 1968 23 90. 1121 MARCHELLO J. M. and TOOR H. L., Ind. Engng Chem. Fundls 1963 2 8. [I31 MEEK R. L. and BAER A. E., A.1.Ch.E. J/l970 16 84 1. D. D., Chem. Engng Sci. 1961 16 287. iI41 PERLMUmER u51 POPOVICH A. T., Ind. Engng Chem. Fundls 1969 8 609. [I61 TOOR H. L. and MARCHELLO J. M., A./.Ch.E. JI 1958 4 97. [,I PI [31 [41

Chemical Engineering Science, 1972, Vol. 27, pp. 442-445.

Pergamon Press.

Printed in Great Britain

A note on the plunging (Received

liquid jet reactor

22 March

PLUNGING liquid jet reactor is defined as the flow geometry formed by a coherent liquid jet plunging through an ambient reactive atmosphere into a bath of the same liquid. The entrainment and subsequent reaction of the atmosphere in the subsurface two phase region generated is the mechanism of interest. The work described herein is a preliminary survey of the effect of nozzle geometry and jet hydrodynamics on the reactor system to provide a basis for planning a more detailed investigation. The potential of the system as a reactor was recognised by Mertes [1] in 1938 when the system was patented as a technique for mixing and reacting liquids and gases. Subsequently this system has been investigated[2,3] but without due regard to the influence of the physical entrainment rate of the plunging jet and its influence on the reactor system. Davies and Ting [4] investigated physical absorption to turbulent free jets and found that the absorption rate increases with Reynolds Number. However the subsurface reaction of the entraining turbulent jet was not considered. Work by McCarthy et al.[5-81 indicates that the physical entrainment rate of an inert atmosphere varies over orders of magnitude essentially as a function of jet stability which acts as the surface shape generator. The controllable system parameters are the nozzle geometry, nozzle height and nozzle velocity. With constant nozzle height and jet exit velocity, a variation in the length of nozzle parallel throat can change the kinetic energy defect of the exit flow due to variation in velocity profile development. This excess energy produced in viscous flow provides an initial destabilising disturbance at the nozzle outlet and the relaxation of the velocity profile can change the jet stability and hence the jet entrainment ratio by orders of magnitude [7]. For a given nozzle design the stability of the jet at a given point will also depend on the jet velocity and the length of the free jet from the nozzle. For any particular liquid, there THE

442

1971)

are therefore three major parameters which describe the jet geometry at the plunge point: viz. the nozzle design, jet length and jet velocity. McCarthy et al. have rationalised the entrainment process in terms of the surface roughness of the jet [7]. They find for a jet of definable surface roughness,

(QG), = (4)s ‘_, QL [ dv 1

(1)

where the geometric variables are defined in Fig. 1 and (QG),/QL is the entrainment ratio for a jet of length X.

Non le

,Jet

ri Jet length X

Jet

ix

surface

envelope

1x

Fig. 1. Specification of jet surface roughness.

Fig. 2. The reactor for two extreme jet conditions. (a). Type A nozzle. Vj = 18-30 msec-‘, c/N = 2*54 x lo3 m, x = 5 dia. (b). Type C nozzle. Vj = 10.70 msec-I, c/N = 2-54 x 10e3 m, x = 75 dia. Exposure 10 psec. flash. Both photos identical field of view and magnification.

(Facing page 442)

Shorter Communications Hence, for the case of jets possessing definable surface roughness, entrainment can be considered as a purely mechanical process, related solely to the scale of subsurface cavities produced by the interaction of jet surface proturberantes with the bath surface. Since McCarthy et al. found that jet surface roughness was the major controlling parameter in the entrainment process, it is likely that the roughness of the jet will affect the bubble size distribution of the two phase subsurface mixture and hence the reaction characteristics of the system. The object of this preliminary work is to investigate this phenomenon. In order to conduct the survey, a system of relatively high liquid phase reaction rate was required so that the subsurface two phase region existed over a finite volume which could be determined by photographic methods. Figs. 2(a) and 2(b) show the plunging liquid jet reactor under two different jet geometry and flow conditions, with the same liquid phase reaction conditions. The reaction was between pure CO, and 2.0N NaOH at 25°C. The figures indicate the large differences in the characteristic bubble size distributions of the two extreme sets of conditions. Figure 3 shows diagrammatically how the two phase mixture can be considered to be a conical volume given by V,=f2

d,‘. h,

also agrees with the work of De Frate and Rush[3]. The reaction rate per unit volume can then be obtained by using the entrainment rate data of McCarthy[5] and photographic measurement techniques to evaluate the reaction volume. It has been shown by Danckwerts [9] that the reaction rate per unit area for the reaction between CO, and 2.0 N NaOH can be expressed by: R=C*fi.‘/&

(3)

provided

and

both of which apply at the sodium hydroxide concentration used and the exposure times involved, which are of the order 10 msec- 1.

(2)

The reaction rate for the reactor can be considered to be equal to the entrainment rate as measured by McCarthy]51 for air/water systems since the entrainment rate,of air and carbon dioxide have been determined for both water and the 2.0 N sodium hydroxide used and shown to be the same. This

2-

0

I

I

I

I

I

I

I

Plunging liquid jet reactor volume

n

2-

I 0

IO

I 20

I 30

I 40

I 50

I 60

I 70

60

Jet length (nozzle diameters)

Fig. 3. Diagramatic

definition of the plunging liquid jet reactor.

Figs. 4-6. Interfacial area related to nozzle design, jet velocity and jet length. Jet velocities 0 0 0 18.30 msec-I, n q 0 15.25 msec-I, A A A 10.70 msec-*, Nozzle diameters 2.54 x 1O-3m.

443

Shorter Communications From Eq. (3) the specific interfacial area is given by Ra a=C*fim

where Ra = reaction rate per unit volume. Using the data of Nijsing, Hendtiksz and Kramers [lo] for C *, D and k2, the specific interfacial area can be evaluated for the reactor. Figures 4, 5 and 6 demonstrate the change in interfacial area wiih nozzle design, free jet length and jet velocity. In general, the interfacial area per unit conical reactor volume is a function of the two parameters, jet surface roughness and jet velocity. This is because the interfacial area increases at shorter jet lengths where the jets are smoother and increases at the higher jet velocities. For all jet conditions except the roughest surface, the plungingjet reactor is potentially a more efficient contactor volumetrically than the typical stirred tank[ll], packed column[l2] or bubble plate[l3] and at the highest values of interfacial area is of the same order as the sonic gas velocity bubble column of Mashelkar and Sharma [141.

In order to rationalise these data, the roughness can be represented by a similar ratio to McCarthy’s[7], dE/dN, and the rate of bubble generation by the velocity of the jet at the nozzle exit. Using these parameters, a relationship of the form:

was tested where:

112 c= I&+, 4 QI, 1 d

Figure 7 shows the correlation between the interfacial area per unit volume of the subsurface two phase mixture and the surface geometry/rate of bubble generation factor. The experimental data can be seen to fit the correlation reasonably well over an order of magnitude range in interfacial area, and the interfacial area bears a linear relationship to the geometry factor over a major portion of the range. The positions A and B on Fig. 7 correspond to the reactor conditions for Figs 2(a) and 2(b) respectively. The differences present in the interfacial area per unit volume for these conditions are due to differences in the bubble size distributions obvious in Figs. 2(a) and 2(b). The correlation of the reaction conditions in terms of the jet geometry parameter has shown that the reaction model for the plunging jet reactor relies significantly on the jet hydrodynamic conditions upstream of the reactor. Further work with a first order reaction will provide the basis for extended testing of the correlation. Acknowledgement-

The authors greatfully acknowledge the assistance of the B.H.P. Co. Ltd., through their Central Research Laboratories during the course of this investigation.

Metallurgy Department The University of Newcastle Newcastle, N.S. W. 2308 Australia

J. M. BURGESS N. A. MOLLOY

B.H.P. Co. Ltd. Central Research Laboratories Shortland, N.S. W. 2307, Australia

M. J. MCCARTHY

[

from Eq. (1).

, a cb: D dE dN d, hr k,

QL R Vr vj Z

Fig. 7. Interfacial area related to jet conditions.

x

NOTATION interfacial area per unit reactor volume, cm-l concentration of liquid phase reactant, kgmole rn+ equilibrium concentration of gas phase reactant in liquid, kgmole m-3 diffusion coefficient of gas phase reactant in liquid, m%ec-* diameter of the gas envelope totally enclosing the jet, m diameter of the jet nozzle, m maximum diameter of the conical reaction volume, m maximum length of conical reaction volume, m second order reaction velocity constant, m3 kgmole-’ set-’ physical absorption mass transfer coefficient, msec-’ volumetric flow rate of gas entrained into the reactor, m%ec-’ volumetric flow rate of liquid in jet, m%ec-I reaction rate per unit area, kgmole m-* reactor volume, m3 jet velocity, msec-* stoichiometric coefficient jet length, m

REFERENCES [I] MERTES A. T., U.S. Patent 2 128 3 11,1938. [2] SWIGGETT G. E., Ph.D. Thesis, Oregon State University, Univ. Micro. Films No. 69-463, Ann Arbor, Mich. 1969. [3] De FRATE L. and RUSH F. E., Gas entrainment into a pool by turbulent liquid jets. Pre print 390, Symp. on Selected Papers-Part II. 64th Nat. Mt., A.Z.Ch.E New Orleans, Louisiana, March 16-20 1969. [4] DAVIES J. T. andTING S. T., Chem. Engng Sci. 1967 2 1939. [5] MCCARTHY M. J., Ph.D. Thesis, University of Newcastle, N.S.W. Australia 197 I.

444

Shorter Communications

[61MCCARTHY M. J., KIRCHNER

W. G., MOLLOY N. A. and HENDERSON

J. B., Trans. Inst. Min. Metall.

1969,

78 C239.

[71 MCCARTHY M. J., HENDERSON J. B. and MOLLOY N. A., Proc. Chemeca 1970. Conf, 1970, Australia. Sec. 2, pp. 86-100. Butterworths. [81 MCCARTHY M. J., HENDERSON J. and MOLLOY N. A., Met. Trans. 1970 13657. 191 DANCKWERTS P. V., Ind. Engng Chem. 1951 43 1460. 1101 NIJSING R. A. T. O., HENDRIKSZ R. H. and KRAMERS M., Chem. Engng Sci. 1959 10 88. P. H., Trans. lnstn Chem. Engrs 1958 36443. 1111 CALDERBANK 1121 DANCKWERTS P. V., RATCLIFF G. A. and RICHARDS G. M., Chem. Engng Sci. 1964 19 325. 1131 McNIEL K. M., Can. J. Chem. Engng 1970 48 252. 1141 MASHELKAR R. A. and SHARMA M. M., Trans. Instn Chem. Engrs 19704fJTl62.

Chemical

Engineering

Science,

1972, Vol. 27. pp. 445-447.

Pergamon Press.

Printed in Great Britain

Numerical evaluation of mean and variance from the Laplace transform 1 June 197 1)

(Received THE CHARACTERISATION of

linear dynamic systems by the moments of the impulse response is a well-established practice. These parameters provide a quite general description of the entire system behaviour without presupposing any particular mathematical model; model matching may then be achieved by equating these measured moments to those obtained from the Laplace transformed model equations using the, by now, well known relationship:

Rearranging Eq. (3) we have:

1 -=-p+q+o(x”); G(s) ; InG(0)

so that a plot of (l/s) In (G (s)/G (0) ) vs. s for small positive real values of s yields a straight line having slope ti/2 and intercept --cc. The mean and variance are thus obtainable directly from the Laplace transform without the need for double differentiation or other mathematical manipulation. (- 1)’ j- ty(t) dt. The only remaining question is over what range of s values Cl the straight line relationship applies: is it sufficiently large for In principle the method is very attractive. In practice it the method to be generally applicable without running into suffers from two limitations: the moments of experimental problems of arithmetic precision in the evaluation of the curves are highly sensitive to small measurement errors, parfunction (l/s) In (G(s)/G(O))? As it happens, these problems ticularly in the long time, tailing, region; and analytical exare unlikely to arise. The following examples illustrate that pressions for the moments from the direct or indirect applifor flow-mixing systems it is only models describing extreme cation of Eq. (1) may be difficult or, indeed, impossible to by-pass effects that could give trouble; in the usual situaarrive at. tions, where the degree of mixing lies between plug-flow and Measurement errors effectively limit the evaluation of perfect mixing, with, perhaps, some tolerable degree of bymoments from experimental response curves to the mean, CL, passing, the method works extremely well, and very few and variance, a2. The purpose of this note is to provide a function evaluations are required. simple method for obtaining these parameters from Laplace transforms when analytical expressions are unavailable. ILLUSTRATIVE EXAMPLES

w =

The dispersion

NUMERICAL EVALUATION OF THE MEAN AND VARIANCE The proposed method is based on the logarithmic expansion of the Laplace transform. Starting with the definition G(s)

= /0mf(t)e-8fdt.

and expanding the exponential term, leads to the general expression for the logarithm of G(s) in terms of the system moments: lnG(s)

model

With realistic boundary conditions, expressions for the transfer function become quite cumbersome. However, our purpose is well served by assuming the normalised form (CL= 1)

= InG(0) -ps+F+o(s”)

(3)

Some implications of this expression to dynamic systems analysis have been explored in a most useful, although apparently little known, paper by Paynter[ 11; and, as discussed in Ref. 121, Eq. (3) often provides a more convenient route for arriving at -analytical expressions for the system moments than the direct application of Eq. (1).

445

(5) which is valid for all P in an open-ended system, and provides a good approximation, regardless of end conditions, for large P. The tanks-in-series

model

This

model provides another illustration of the method. In normahsed form the transfer function is: G(s)=

(

;+1

-n >