Chemical Engineering Science, 1968, Vol. 23, pp. 783789.
Pergamon Press.
Printed in Great Britain.
Mixing characteristics of a bounded turbulent jet T. WOOD Department of Chemical Engineering, The University of Sydney, Sydney, N.S.W., Australia (Received 29 November 1967) AbstractThis paper considers the problem of predicting jet of water flowing horizontally through a cylindrical or and exit. An approximate analysis, based on the known expression which is tested experimentally for twentyone INTRODUCTION
work represents an attempt to predict the mixing characteristics of a flow system from a knowledge of its flow field. The concepts needed to describe mixing in a flow system were first defined by Danckwerts[ 11 and later added to by Zwietering[2]. Thus the terms “residence time distribution”, “age distribution” and “life expectation distribution” are in common use and their relationships to mixing and to each other are wellknown. Levenspiel[3] provides a useful summary of references relating to work in this field. The mixing in a flow system is determined by the flow rate and properties of the fluid, and by the geometry of the system. The latter comprises the size and shape of the vessel and the locations and sizes of the inlets to, and the exits from, the vessel. The author has pointed out in a previous communication that in certain cases of ideal fluid flow the residence time distribution can be determined from first principles [4]. The solutions to such cases, though, are necessarily limited in their application and are mainly useful in defining limits to the behaviour of real fluids. This paper considers the problem of predicting the mixing caused by a bounded turbulent jet, resulting from a steady flow of water through a cylindrical or rectangular vesse1 with axially opposed inlet and exit. An approximate analysis, based on the known properties of a free turbulent jet, leads to an expression for mixing, which is tested experimentally for several vessels. An THIS
the amount of mixing caused by a turbulent rectangular vessel with axially opposed inlet properties of a free turbulent jet, leads to an different vessel geometries.
assessment is then made of the conditions which the analysis is valid. MATHEMATICAL
for
ANALYSIS
Description ofJowfield Figure 1 illustrates the simplified flow field. Two distinct regions are postulated; one consisting of the jet itself and the other, for con‘surroundings’. Visual venience, designated observation with a dye tracer suggested this model and also confirmed the presence of a nonturbulent reverse flow in the surroundings. The velocity profile in each region is assumed to be ‘tophat’ in form and the jet region is assumed to have similar characteristics to those of a free turbulent jet: in particular, the boundary is linear and the axial volumetric flowrate increases linearly with distance from the inlet[S,6]. In consequence, fluid is entrained from the surroundings into the jet region and this mechanism is responsible for the mixing produced in the system. The jet properties, developed on the basis of constant momentum rate as in a free jet, can be summarized as follows:
783
u =
UO
$= 0
(
1+;.tae
I+;.
tan 8
1 l
(lb)
T. WOOD
h Inlet
___c
0
Jet reaion
1
exit

A
t t t
Entrainment

(a)
Reverse
1
0
flow
z L? 0
Fig. I (a). Simplified flow field showing actual size ofjet region.
Surrounhngs
Area
vebdty
Aa
”
. 
Coxentmticn c Area a E Lx Entmhment rate
x=0
xt8x
x
(b)
Fig. l(b). Enlarged section for development of Eqs. (2a) and (2b).
ac E ac at+U.a,+a.c=&s
where r is the radius of the jet, ic is the velocity in the jet, Q is the voIume flow rate in the jet, x is the axial distance from the inlet, tan 8 is the jet halfangle.
Development of the Ffunction for the system Consider the response of the system to a step change in a tracer. Then a mass balance on each leads to the following region, separately, equations for the jet and the surroundings:
$+,.$)
(Jet) (W (Surroundings)
(2b)
where c. s are the tracer concentrations in the jet and the surroundings respectively, U, v are the velocities in the jet and the surroundings respectively, E is the volume rate of entrainment into the jet region per unit length, a is the cross sectional area of the jet. t is time measured from 784
Mixing characteristics
the step change, x is the axial distance from the inlet. For a step change in tracer concentration from 0 to C, the following initial and boundary conditions apply: t=O t>O t30
OSXCL X= 0 x= L
c=s=o c=c c=s
where L is the length of the vessel. The solution to the equations (Appendix A) gives the following expression for the F function: * H(tt,*)
(3)
where K is the ratio, total entrainment rate into jet total volume rate in jet at x = L tt is the time required for an element of fluid to travel from x = 0 to x = L; T is the holding time, i.e. vessel volume inlet flow rate H(tt,*)
= 0,
if
0 S t G tt”
H(tt,*)
= 1,
if
t > tf.
of a bounded turbulent jet EXPERIMENTAL
The vessels used in the experiments were made up in Perspex and each vessel could be fitted with a series of interchangeable inlet and exit nozzles: the inlet and exit were axially opposite one another (Fig. 1). In an experiment, a steady flow of diluterNaC1 solution (approximately 2 g per litre to minimize density effects) was admitted to a vessel. At a certain instant, a step change to pure water took place in the inlet stream. The effluent from the vessel was then collected in a series of flasks over known, discrete time intervals. The NaCl content of each flask, determined by titration, enabled a cumulative distribution of ‘NaCl disfrom the step placed’ vs. ‘time measured change’ to be determined. In this way, the function J, in Eq. (5) was estimated for each vessel at a number of values oft.
. 1 RESULTS
From the set of experimental $I t values for each vessel, a test of Eq. (5) was made by plotting the semilogarithmic relationship between (1  JI) and the nondimensional time t/T. Figure 2 shows I.0
Relationship between the Ffunction and experimental measurements The experiment is designed to measure $, the fraction of material present in the system at some instant t = 0, which is displaced from the vessel in the interval from t = 0 to some subsequent instant t. J, is, in fact, Zwietering’s life expectation function[2] and is readily shown to be related to the Ffunction by: z=$F). Now F is given by Eq. (3) and, as shown in Appendix 3 and Table 2, f: is negligible for the systems studied in the experiments: thus the following expression can be deduced from Eqs. (3) and (4):
030
\
\
(5)
.
\
t
\
om
*
I
‘*Y
o..IL+_ t 0
2
3
v
Fig. 2. Experimental
JI= 1expKg.
.
results for 4 in. cube with &in. inlet
T. WOOD
a typical plot of this relationship for the 4 in. cube. The linear relationship was confirmed for each of the vessels studied and the value of K determined by a ‘least squares’ regression analysis. A series of preliminary experiments showed that K was solely dependent on vessel geometry, and not on flow rate, provided that the jet was fully turbulent (Reynolds number, based on inlet dia. greater than 2000). Values of K for each of the vessels studied are summarized in Table 1. K, previously defined in Eq. (3) and Appendix A, is given by:
angle, can be computed. They are summarized in Table 2. The mean value of 8 for the vessels studied is 6.3”. Figure 3 shows a comparison of experimental JI values and $ values calculated from Eq. (5) and (6b) for vessel D, the 4 in. cube, using a value for 8 of 6.3”. Table 2. Values of 0 and tL*/r for each vessel Vessel A
K=+?.
Length/dia 1.04
:
(6a)
L
Inlet dia. (in.)
a
tL*/Tx 103
e”
1.4 2.8 6.0
6.2 6.0 5.2
Thus, using Eq. (1 c): B
1+~.cote
K= (
1. >
From Eq. (6b) and the values of K deduced from the experiments, values of 8, the jet halfTable 1. Values of K in Eq. (1)
Vessel A
B
C
D
Type Cylinder
Cylinder
Cylinder
Cube 4 X 4 X 4(in.)
Dimensions (in.) D = 5.75 L=6
K
2 a
0.95 0.91 0.82
Cube 8 X 8 X 8(in.)
0.84
a
0.93 0.86 o77
D = 575 L=4
tDE=4 L=4
Is
DE=8 L=8
$
a F
G
Box 4 x 4 x 8(in.) Box 4 x 4 x 8(in.)
c
E
DE = 5.33 L=4 D,=4 L=8
0.94 O86 0.76
a $
O98 o94
a
a
1.6 3.2 7.9
6.4 5.6 5.6
O6 I.4 3.5
5.1 5.8 6.3
o9
4
0.75
200
4 a
::;
6.8 4.9 5.4
&G
4.0
9o
:
16.3 7.2
7.5 77
DISCUSSION
The theory developed in this paper is based on the premise that free turbulent jet behaviour is valid for the vessels studied. The consequences of this will now be examined. The free turbulent jet model does lead to an expression that is consistent with the experimentally observed relationship between JI and t. Furthermore, Eqs. (5) and (6b) predict, for a given flow rate, that as the inlet diameter.to a vessel decreases the entrainment rate increases and so the amount of mixing increases. In the limit, as the inlet diameter becomes very small,
o”‘z O86 . 0.94 O85 0.75
*
G
0.7 :I;
t F
::: 6.4 5.6 5.4 6.1
1.00
1.00
140.; 218
; t
0.70
D = 3.25 L=6 a
& :
D
Inlet and exit dia. (in.)
: E
1.85
(6b)
090
tDE is the hydraulic diameter.
786
Mixing characteristics
0
Llb
l
& in.clia inlel
of a bounded turbulent jet
indii.
in.&. hht +$\bkaa pmdickd fmn
!cqs.(5) md (6b)
using8=6.J0
oa_I 0
2 KI
Fig. 3. Comparison
3
T
of predicted and measured values for 4 in. cube.
the mixing becomes complete (in the sense that the JIt relationship becomes that of the socalled ‘perfectly stirred’ system). Also, values of 8 computed from Eq. (6b) are consistent with those quoted in the literature for a free turbulent water jet [7,8]. The literature values vary over the range 7”+: l$‘, while the experimental values from this study have a mean of 6.3” and a standard deviation of 1”. The largest differences from the mean are associated with the two vessels B and G, (Table 2), whose L/D ratios are largest. The clearance between the jet boundary and the wall becomes less as L/D increases (Fig. I), velocities in the reverse flow region become appreciable and the freejet theory is less appropriate. Against the freejet model is its failure to describe conditions in the region of the exit from the vessel. Clearly, the recirculation of entrained fluid and the contraction of fluid towards the exit nozzle create a completely different flow situation to that shown in Fig. 1. Some recent work, though, has shown that this
region is quite small in relation to the total volume[9], and that the conditions shown in Fig. 1 hold for most of the vessel. The mixing effect is caused by entrainment from the surroundings into the jet region along the entire length of the vessel and the experimental work seems to confirm that the influence of the ‘end effect’ is small. On balance, then, the freejet model seems to be adequate to describe the mixing caused by a turbulent jet in vessels whose geometry is such that the velocity in the reverse flow region is always small compared to the velocity in the jet region. The practical applications of jet mixing are, of course, well known: for example, in ejectors, in flame combustion and in buoyant plumes from chimneys. One further example is particularly worth noting: Fossett[ lo] reported a difficult problem in blending components to aviation gasoline stored in large inaccessible underground tanks. The solution was finally achieved by the creation of a turbulent jet in the tanks. It remains true, though, that the design of vessels used in continuous processing often does not take into account the consequences of,such factors as the influence of inlet (and exit) location and size on internal mixing or the extent to which baffles or stirrers are often prescribed when they are not needed. These sorts of problems might well be worthy of further study. CONCLUSIONS
The mixing caused by a turbulent jet of water flowing horizontally through a vessel of the kind shown in Fig. 1 can be predicted from the following equations: $I= 1expKi K=
(If?.
[email protected]’
where 8 = 6.3” (for water). The analysis is based on the assumption that the vessel geometry satisfies the following condition
787
T. WOOD
R vessel radius t time t* time for particle to travel from x = 0 to x U velocity in jet velocity in surroundings ; vessel volume x axial distance from inlet e halfangle of jet holding time ; lifeexpectation function
NOTATION
a jet area concentration of tracer in jet region d concentration . of tracer in inlet stream (t = 0) d jet diameter D vessel diameter E entrainment rate per unit length of jet F residence time function H Heaviside’s step function K mixing parameter in Eqs. (3) and (5) L vessel length Laplace transform variable volume rate in jet e” r jet radius
Subscripts 0 condition at x = 0 L condition at x = L
REFERENCES
111DANCKWERTS, P. V., Chem. Engng Sci. 1953 2 1.
PI ZWIETERING Th. N., Chem. Engng Sci. 1959 11 I. [31 LEVENSPIEL O., Chemical Reaction Engineering. Wiley 1962. 196217391. [41 WOODT.,Chem.EngngSci. RICOU F. P. andSPALDING D. B.J. FluidMech. 19611121. tz; MORTON B. R.,J. FluidMech. 196110 101. 171 MOSS E. T.. Trans. Instn them. Enws. 1947 25 191. Instn them. Engrs. 1959 37 255 181 DONALD M. B. and SINGER H.,kms. 191 PHILLIP R. G., M. Engng SC. Thesis, University of Sydney 1966. [lOI FOSSETT H., Trans. Instn &em. Engrs. 195 129 323.
Elimination the boundary
APPENDIX A Laplace transform analysis Taking transform of Eqs. (2a) and (2b), and rearranging, gives:
of the constants I, and Z2, using conditions at x = 0 and x = L,
gives:
0c L=p00 c
(24 ’
exp_pTL
xQ~.expp(t,*T~)f(~,L)f(~,O)’
(2b)’
z+,.a=o.
(2) ‘I’ Determination off(p, L)  f(p, 0)
Solutions to (2a)’ and (2b)’ are: E= exp;pt*
{II + 12 *f(p, x)}
f(p,L)f(p,O)
(,a)”
But
S = Z2* exppT
(t*T)
where f(p,x)
=IE*
expp(t*
=~!iE.expp(t*T)
T)  dx
=I
(~$k
AQaQo s= I QCQQo>
* **
Now, ifAQ 9 aQo, then: I*T=$lnx
and I,, Z2are integration constants. 788
(since,QQo=Ex).
‘dx.
Mixing characteristics
of a bounded turbulent jet
17,F . dt = I: (1  F) dt
Therefore expp(t*
(Holdback criterion).
T) = xdA’=Q But, since tz is negligible in the present experiments, then the Ffunction becomes
and .
_ 0 b
F= IK*expKm:.
Substituting in (2)“’ and rearranging, gives: c
c L=
1 p+
K (k,7)
exp_pt,*
I
APPENDIX B The velocity in the jet region is given by
where K=
QLQo Qo
Since the Ffunction
uo
'
is given by (c/C), then K(tt,*) 7
(7_tz)
t,
03:
Ldx 
Iu 0
L/. rttan8 L .
12)
=G
This solution has the properties t = 0;
.
1
* H(t12). tt*=
K(t
1
(1B)
The time required for a fluid particle to travel fromx=Otox=Lisgivenby
A more precise solution would be: IK.exp
(\ l+$.tan8 ro
&=
In nondimensional
I
\
4
>
form
F=O F1
R&um& Ces expose considbre le problbme de prediction de la quantite de melange cause par un jet turbulent d’eau s’ecoulant horizontalement a travers un mcipient cylindrique ou rectangulaire, avec une entree et une sortie oppodes par rapport P I’axe. LJne analyse approximative, bade sur les prop&t& connues d’un jet turbulent libre, conduit B une expression dont on fait I’essai exp&imentalement pour vingtetune caracttristiques geometriques differentes de recipients. ZusammenfaasungIn diesem Artikel wird das Problem der Vorhersage der Mischwirkung eines turbulenten WasserstrahIes behandelt, der sich horizontal durch einen zylindrischen oder rechteckigen Betilter mit axial gegeniiberstehendem Eingang und Ausgang hindurchbewegt. Eine RohAnalyse auf der Grundlage der bekannten Eigenschaften eines freien turbulenten Strahles IIIhrt zu einem Ausdruck, der experimentell fiir einundzwanzig verschiedene Behiterformen gepriift wird.
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