References
(1) Anderson. G. H.. Haselden, G. G., Mantzouranis, B. G., Chrni. Eng. Sci. 16, 222 (1961). (2) Ibzd., 17, -51 (1962). (3) Brnnett. J . A. R., Collier, J. G., Pratt, H . R. C., Thornton, .J. D., 7 r a n s . Inst. Chem. Engrs. (London) 39, 113 (1961). (4) Chen, J. C.. “Correlation for Boiling Heat Transfer to Satu-
rated Fluids in Convective Flow,” A.S.M.E.-A.1.Ch.E. Heat Transfer Conference, Boston, 1963. (5) Collier. J. G., .4t. Energy Res. Estab. (G. Brit.), Rept. AERE CEl’R 2496 (1 957). (6) David. M. M., “Literature Survey on Boiling in Closed Channrl Flow Systems at High Vapor Fractions,” report to Roeing .4irplane Co., Seattle, Wash., 1957. ( 7 ) Davis, E. J.. Ph.D. thesis, University of Washington, 1960. 181 13avis. E. J.. David. M. hi.. Can. J . Chem. EnP. 39. 99 (1961). 19) (9) Deissler, R. ’G.> G.. Natl. Advisory Comm. Aeronaut. Tech: Notis 2129 (1950), 2138 (1952), 3145 (1959). (10) Dengler. C . E E. . Ph.D. thesis, Mass. Institute of Technology, ,logy, 1.(l6? ,
(1 1) DrngIer. C. E.. Addoms, J. N., Chem. Eng. Progr. SyTc2p. Ser. 52, 95 (1956). (12) I h k l r r . A. E.. Ibid.,5 6 , 1 (1959). (13) Dukler. A. E.. Ph.D. thesis, University of Delaware, 1951. (141 Iluklrr, A . E.. Bergelin, 0. P., Chem. Eng. Progr. 48, 557 (1952). (15) Fikr)-, M. M.. Ph.D. thesis, Imperial College, London, 1953. (16) Forstrr. H. K., Zuber, N., A.I.CI2.E. J . 1, 531 (1955). (17) Groothuis. €I.! Hrndal, 1%‘. P., Chem. Eng. Sci. 11, 212 (1959).
(18) Guerrieri, S. A , , Talty, R. D., Chem. Eng. Progr. Symp. Ser. 52. 69 (19.56). - -,(15) Hewitt. G. F.. At. Energy Res. Estab. (G. Brit.) Rept. AERE-R-3680 (1961). (20) Hsu, Y . Y . . Trans. A.S.lM.E . 84, 207 (1962). (21) Knuth, E. L.. “Evaporation fr’om Liquid’ \Val1 Films into a Turbulent Gas Stream,” p. 173, Heat Transfer and Fluid Mechanics Institute. Stanford Univ., 1953. (22) Kvamme, A , , M.S. thesis, University of Minnesota, 1959. (23) Lee, G.? M.S. Thesis, Mass. Institute of Technology, 1952. (24) Lockhart, R. \V.. Martinelli. R. C.. ChPm. Enp.. Progr. 45, 39 (1949). (25) McAdams, \V. H., “Heat Transmission.” 3rd ed., McGrawHill, New York, 1954. (26) Mumm, J. F.. Argonne Natl. Lab. Rept. ANL-5276 (1954). (27) Ibid., BNL-2446 (1955). (28) Parker, J. D., Grosh, R . J., Ibid., ANL-6291 (1961). (29) Sani, R. L., UCRL Rept. UCRL 9023 (1960). (30) Shrock, V. E., Grossman, L. M.:Suclear Sci. Eng. 12, 474 (1962). (31) Silvestri, M.: Finzi, S..Roseo. L.. Schiavon. 14..Zavattarelli, Z., Proc. Second U S Intern. Conf. Peacqiul 17resA t . Energy 7 (1958). (32) Vanderwater, R. G.: Ph.D. thesis, Lniversity of Minnesota, 1957. (33) van Rossum, J. J., Chem. En,g. Scz. 11, 35 (1959). (34) Warner, C. F.. Reese, B. A , , Jet Propulsion 27, 877 (1957). 7
\ - -
RECEIVED for rei-iew April 24, 1963 ACCEPTED October 5. 1963
PROFILE RELAXATION IN NEWTONIAN JETS STA NLEY M I DD L EMA N
,
Ciizaeiszty of RochestPr, Rochester, S. Y .
An approximate mathematical model i s formulated from which the decaying velocity profile, and the diameter, of a laminar Newtonian jet ejected into air from a long circular tube may b e obtained as a function o f distance from the tube exit. From this solution, the distance required for the diameter t o become almost constant may b e predicted. The theoretical results a r e in good agreement with experimental observations.
HE. LIQUID JET is a valuable research tool in investigations Tof interest to the chemist and chemical engineer. Because the jet presents a continually renewed and relatively pure surface. it has found use as a system for the study of surface tension ( 7 : 9 ) . Because its surface is usually well defined geometrically. it lends itself to the measurement of the kinetics of ahsorption and reaction a t liquid surfaces (4, 70). The Ftability of liquid jets has received much attention lately because of its importance as a controlling factor in some rocket combustion systems ( 8 ) . >fore recently. liquid jets have been used in fundamental rhrological studies, particularly in attempts to develop techniques for the measurement of normal stresses in polymer solutions (2. 5, 71, and in attempts to measure stress relaxation in polymer solutions ( 3 , 6 ) . ‘[he primary virtue of the liquid jet lies in the fact that it projects a very short lifetime (the order of 1W2second) ovrr a relatively large space (the order of one foot) thus lending itself to the study of such rapid processes as absorption and stress relaxation in liquids. In most situations. the jet is formed by ejecting fluid from an orifice or tube. T h e fluid leaves with a nonuniform \’?locity profile. and, if ejection is into a n ”inviscid” medium Filch as air. this profile is free to relax to uniformity. T h e rclxuarion process gives rise to a number of effects. I n lx~rtiriilar.rhe surface velocity differs from the average bulk
118
I&EC
FUNDAMENTALS
velocity of fluid. Hence, if the average velocity is used to calculate the “age” of the jet? this age is different from the surface age. Since the surface age must be known in order to measure, for example. the rate of absorption a t the surface. one must either minimize the effect. or! as in the work of Hansen (4). account for it by Eome correction factor. Profile relaxation is generally accompanied by a slight diameter change of the jet ( 7 ) . Since surface area is often of prime importance. it is essential that this diameter variation be recognized and accounted for. Finally. in the viscous and viscoelastic fluids of rheological interest, profile relaxation gives rise to recondar)- normal stresses. LVhether these stresses are of a magnitude comparable to the stresses of primary interest is, a t present. an unanskvered question. Bohr ( I ) recognized the existence of a nonuniform velocity profile and developed an estimate for its rate of relaxation. H e concluded that relaxation occurred ivithin a short distance of the jet exit. Rut irone is considering dynamic phenomena---such as stress rrlaxation or the growth of surface disturbancesthen these phenomena occur primarily in the exit region. and the possibility of intrraction \\-ith profile relaxation must be considered. The points raised in this discussion have prompted examination in some drtail of rhe dynamics of viscous jets in the exit region. I his paper is concerned xvith the problem ~
Conservation of mass can be expressed by:
of profile relaxation in a jet of Seivtonian liquid emerging in fully developed laminar flow from a long circular tube. ,4n approximate solution for the velocity profile has been obtained, and quantities derived therefrom are compared with experimental results.
As a boundary condition. one states that no shear stress is
Theory
exerted on the jet surface. If the radial velocity component is neglected, the boundary condition is
T h e radial and axial velocity components, U, and u z , and the pressure. P? are determined from the Navier-Stokes equations with the equation of continuity. For a flow \cith axial symmetry, these equations are
L'
resds
'The neglect of radial velocity in this expression will be justified after examining the experimental results. .4s an initial condition, L L ~ is taken to be the fully developed laminar profile : 2(1
+ 1r bra
- -
(ru,) = C
(3) = a({)
~e,
N'hile the nonlinearity of the equations is sufficient to prevent one from obtaining their solutions, the \\-orst feature of the equations appears upon \vriting the boundary conditions. for the radius of the jet. r 3 . does not remain constant, and is unkno\$n. 'This leads us to seek a n approximate solution along the following lines. First: dimensionless quantities are introduced : _-
U ) = L',
Re
=
:v
(4)
s
=
2r./do
(5)
r
=
Re z/dg
(6)
2rj
do
= vdop/'p
re2sds
=
b
(8)
Cl
.-A more rigorous discussion of the momentum balance for a jet is available in an earlier publication ( 7 ) . For jet trajectories Lchich are nearly horizontal. gravitational terms can be omitted. .A more serious approximation is the neglect of the contrihution of certain viscous stresses in the momentum balance. 'The rigorous momentum balance including viscous stresses proves to be intractable practically. But earlier \vork [7) indicates that for Reynolds numbers above 100, the contribution of viscous stresses is negligible compared u i t h the contribution of the inertial terms accounted for in Equation 9. Hence. a theory based on this simplified momentum balance cannot he valid for Reynolds numbers much below 100. Surface tension has been shown (7) to be a n important factor in the behavior of low speed jets. But: for all practical experiments. jets of Reynolds number greater than 100 are of su6ciently high speed that the contribution of surface tension mav be ignored. Finallv. \ye remark that if the initial velocity profile is knoivn. the constant C1of Equation 9 can be obtained.
(13 )
S o w Equations 9 through 11 must be used to evaluate the unknoicn functions a? 6 . and c. Equation 13 is substituted into Equations 9 throuqh 11, with C1 and C2 evaluated from Equation 13. Since the upper limit of integration i n Equations 9 and I O . and the hounda~y condition in 1 1 . involve thr unkno\\n 6. t h r w rirririltarieous algebraic equations are otitnincd in the functions a . b. a n d c, \\-it116 appearing expliciti?.. The solutions for a , 6. and care simpl\. ohtained. and give 1
62
=
-
(7)
'I'he axial velocity component must satiefy the macroscopic equation of conservation of axial momentum:
1
+ 6({)s2 + c(r)s'
S o t e that ic is an even function ofs
a =
Reynolds number
6
(12)
- SZ)
Equations 9 through 11 impose three conditions on re(!, 0 % while Equation 12 allows C1 and C2 to be evaluated upon setting 6 = 1 in Equations 9 and 10. Xow it is assumed that the velocity. it. can be expressed as a polynomial in s, Liith coefficients ivhich are functions of the axial variable :
and
bz
(10)
=
8=, ) : : (
26' =
bv,
Cz
=
and
c =
+ d1j~ ' 3 6 2
(14)
\/fi
(1 .i)
/) 'J4
4 G 11 766
(16)
T h e problem is no\\- reduced to finding 6 as a function of {. Thi? will be done b\- :\-orking \\-ith Equation 2. In essence. Equation 2 \ T i l l be \\-ritten along the center line of the jet (s = 0):and its is assumed to satisfy this equation. 'I'he x s u l t Mill be a differential equation for 6. .\long the center line, ci =
0
(181
and Equation 2 becomes
To this point, the development has follo\\ed along the liner or a boundar>- layer analysis. in \vhich the effect of a small radial velocity on the development of a n axial profile i r nrglected. Consistent Ivith the usual boundary layer approximations then. \\-e aswme that the pressure variation is unimportant. and bP a z is neglected compared \\ith the other terms in Equation 1 9 . I n dimensionless variables. then. Equation 19 becomes. VOL. 3
NO. 2
M A Y
1964
119
six constants A and B. These constants are determined from conditions in Equations 32 and 34, as well as four additional conditions which match the solutions, and their slopes, a t the interior boundaries of the three regions. Thus the boundary conditions become, with reference to Figure 1,
From Equation 13, it can be shown that
(i:
g)r=o
+f
= 46
da
= a
Pl(0) = 1
(34a)
p3(a) = 0
(340
aa'
-
d(
w($)r=~
d2a
and Equation 20 then becomes
a" - aa' = 16b/ReZ
(24)
After substituting Equations 14 and 15 for a and b, a n ordinary differential equation is obtained for 6() :
r1
where the coefficients are functions of 6. I t is more convenient to rewrite this equation w i t h p (Equation 17) as dependent variable. In addition, the square of d6/dj- is neglected in Equation 25. T h e result is
Obviously one must make some choice for and [z. I t is observed experimentally that the radius change is nearly complete within about three diameters of the tube exit. We shall take (Equation 6)
and
{Z
= 3Re
(35)
rl
= Re
(36)
As a rough estimate, we take,
2: (27)
+ 7.7 pZ + 10.7 p3 4 -
5.7 - 10.4 p
Q= and 6i
=
-
1800 ~
Re2
(28)
~
p 4 3 +p2
A solution to Equation 26 is now sought. a t ( = 0,6= 1, and hence
P(0)
< 6 < 1.0 and b, = 0.95 0.88 < 6 < 0.96 p = 0.50 4 5 / 2 < 6 < 0.88 p = 0.25
for region 1 : 0.96
where the coefficients are given by
= 1
From Equations 27 to 29, the coefficients, 6, Q,and (R may be evaluated for each region. and from them, CY and P may be determined for each region. Then Equations 34a through 34f allow A i and Bi to be determined. This tedious, but otherwise straightforward, algebraic work has been carried out.
I t is known that, Results
(30)
In addition, as j- -+ 0 3 , the velocity profile flattens and becomes a constant, independent of s. From Equation 13, this means that b(w) = c(w) = 0
3:
Only the solutions for the initial and final regions are needed in the following material. With good approximation, p in the initial region is given by
pl(6) = eP1r
~r
(31)
I
dmD1
r
Equations 15 and 16 show that Equation 31 holds if, and only i f , p = 0. Hence, the second boundary condition is
(37)
~~
' (38) ^ . 1
p(w)
=
0
(32)
From Equation 17, this implies 6 = 4 3 / 2 a t ( + m . This result has been shown theoretically (7), and confirmed experimentally, so long as R e > 190. As the final step, Equation 26 is solved by splitting the r-interval into three regions, as in Figure 1, over each of which the coefficients 6, Q, and (R may be taken as constant. T h e result in a set of three linear ordinary differential equations, for which the general solutions can be written immediately. 'The solution over the z-th region is given by
pi
=
Atead
+ B&%T
(33)
where CY and are knobvn functions of 6, Q, (R, and Re. Since there are three equations of the form of Equation 33, there are 120
l&EC
FUNDAMENTALS
J
and a1 = -
-
26
0.75 [1
+d l +
E]
(41)
02 0
:,
I.o
05
0
3Rs
R8
l/do
L
Figure 1 . Three-region model for solution of Figure 3. The ratio, 4, of surface velocity Figure 4. Comparison of the initial to average velocity, as a function of the condition satisfied by the model, Equation 26 Equation 13, with a parabolic pronumber of diameters from the tube exit 6 voriotion assumed. p curve from Equotion 17. p i , file p:, a n d p j token as characteristic values for p over each region, from which the coefficients in Equation 26 a r e evaluated
Decay Length. -4 measurable property is the distance, j-*. required to reduce the jet diameter to within 1Yc of its final value. This means (3
= {3*
= 1.012/3
'2
(44)
and, from Equation 17, p3* 0.24 (45) Fubstituting Equation 45 for p 3 in Equation 39. and solving for (. one obtains j-* as a function of R e . .l'he results are plotted in Figure 2. where, instead of {*. the parameter Z * do. which represents the number of tube diameters required for 99Yc decay. is Fho\vn. Experimental results--discussed below--are shown on the same plot. Surface Velocity. . I s mentioned earlier. the ';surface age" of the jet differs from the "age" that would be calculated from the ratio of the distance of a point from the exit, and the mass average velocity { z ' ~of ) the jet. This follows from the fact that a particle on the boundary must accelerate from zero w1ocit)- (assuming a "no slip" condition to hold at the exit) to the final jet velocity. and hence \vould be expected to lag behind the mass average velocity. T h e rate at \vhich the furface velocity approaches the average jet velocity can now be calculated. From Equation 13, the surface velocity is given by
zes6 = 100,
'2= a + b62 + ~6~ V
(46)
,
Re =qdop/p
Figure 2. Number of diameters downstream required to reduce the jet diameter to within 1 % of its final value, os a function of Reynolds number D a t o o b t a i n e d using solutions of glycerol-water
\Vith Equations 1 4 through 16. this becomes
Since the parameter of interest here is the ratio of surface velocity to .jet velocity. evaluated a t the same downstream position. the mass average velocity (zsz) is introduced. T h e macroscopic continuity equation shows that
v
(48)
= 62(0,)
from which
4
=
Llz6~(uzj = 1 - 4 1 5 ~ 1 6
(49)
Sotice that a t the exit. p = 1! 6 # 0. This is incorrect. and is a consequence of the approximate velocity profile assumed. Application of Equation 37 gives 0 = 1 - (4Gi6)e'"
(50)
Figure 3 shows the approach of 6 to unity as a function of z , do, using R e as a parameter. Discussion of Results
By the approximate nature of this analysis, it has not been possible to require that the velocity profile satisfy all of the boundary and initial conditions ivhich should be imposed on the Savier-Stokes equations. For example. no attempt \\as made to make Equation 13, the assumed velocity profile, satisfy the initial condition of Equation 12. I n Figure 4. the initial profiles are compared and one notices immediately that a "no slip" condition is violated a t the tube exit. I n addition. the profile shapes differ someivhat. particularly near the boundary of the flokv. O n e would expect profile shape. rarher than magnitude. to control relaxation. since the velocity gradient. and not the magnitude. determinrs the flux of momentum \vhich gives rise to decay. At this stage. ho\iever. \ve are not motivated to seek an improvemenr of the form assumed in Equation 13. I n \vriting Equation 9. the macroscopic momentum balance. the viscous stresses tvhich accompan)- profile relaxation have been neglected. T o include them. ho\vever. \vould be to losr the simplicity which results in Equations 14 through 16 for a . b . and c. T h e approximation is reasonable so loris as the final jet diameter ratio is not much larger than dj 2 = 0.866. For this reason. no validity of the results is claimed [or R e < 80. for \\-hich the final diameter ratio is kno\vn ro escecd 0.00 i/l. VOL. 3
NO. 2
MAY
1964
121
Decay Length. I n Figure 2 the predicted decay length. taken as z* do. is compared \+ith the measured decay length. T h e experiments were performed as discussed in detail elsewhere ( 7 ) , and experimental aspects shall not be discussed here. T h e agreement between theory and experiment is quite good. T o some extent. this result is controlled by the choice of {* = 3Re (Equation 35). for it could be shown that z* do must approach the value 3 for very large Re. O n the other hand, the general shape of the curve is not controlled by this choice, and is in good agreement with the observed variation of z*/do with Re. As mentioned earlier, Bohr developed a n estimate for the decay length of a jet. H e did so by linearizing the SavierStokes equations and achieved this by assuming that the jet radius remained constant. Thus, his theory would not be valid, except possibly near a Reynolds number of 16, for which it is observed ( 7 ) the radius does indeed remain constant. O n Figure 2, Bohr's result is shown, a n d one observes that it crosses the experimental interpolation near R e = 20. in agreement with this conjecture of the validity of his result. No data are shown in the range 10 < R e < 100, as in this region the jet diameter does not vary greatly, and hence the point a t which the final diameter is reached is extremely difficult to measure. I n formulating the boundary condition stated in Equation 11 it was implied that
that the velocity and diameter would equilibrate over roughly the same distance. This discrepancy must be attributed to the approximate nature of the solution. T h e initial period of these curves must be incorrect. since, from Equation 50. q ( 0 ) = 0.36. rather than zero. However, the effect of this shift becomes negligible rapidly, a n d does not contribute greatly to the rapid approach of 6 to unity. I n the absence of any experimental data on surface velocities, it is impossible to reach any definite conclusion about the validity of Figure 3. Acknowledg rnent
T h e author acknowledges the assistance of A. M. Vossler, who obtained and correlated the data on jet diameter decay. Nomenclature
r
r5 S UT
02
ka) V W
-
v
.
z -
z
Re
Yj
For v i , we take the ratio of the change in jet radius, hrj, to the time t* required to achieve 99% of this change. At most z
= constants defined by Equations
P P 6 ,0,a
As a rough order of magnitude estimate, we can take
Arj
C,, C* do
> > bz -
bU, br
= integration constants in Equation 33 = dimensionless functions in the profile model,
a, @,, e ,
bv,
bv, br
A, B a, b , 6
0.147,
(53)
a, P
6
r
P
P t* zz
z*/8
(54)
z
d
3 do
1, 2, 3 refers to quantities evaluated in the corresponding regions of solution of Equation
6
refers to quantities evaluated a t the jet boundary, s = 6
26
SUPERSCRIPT * refers to quantities evaluated a t a n axial position such that 6 = 1.01 4 1 2
As a condition for the approximation, we find -
-rTi, > > -0.14 -9 v doa ~j
Literature Cited
= doj 1
>> 0.14/36
(58)
This clearly justifies the approximation. Surface Velocity, Figure 3 shows that the surface velocity approaches the mass average velocity very rapidly, in all practical cases reaching 99% of the average velocity well within a distance of one diameter. This would indicate that the profile flattens more rapidly than the accompanying change in jet diameter, which requires t\vo to three ,jet diameters for equilibrium. While it is possible that the average velocitv. and hence the jet diameter, continue to change after one diameter, with the surface velocity folloiving so closely behind the average velocity that o z 1. it seems more likelv 122
coefficients in the solution, Equation 33 dimensionless jet radius dimensionless axial variable fluid viscosity 3.1416 = fluid density = ratio of surface velocity to average velocity
= = = = =
2
Hence
or, since 2rj
initial jet diameter pressure defined by Equation 17 coefficients in the differential Equation 26 radial variable jet radius = dimensionless radial variable = radial velocity = axial velocity = mass average axial velocity = mass average axial velocity a t { = 0 = dimensionless axial velocity = axial variable = Reynolds number = doVp/P
= = = = = =
SUBSCRIPTS
a n d , for the data presented here Z*
9 and 10
= coefficients in the differential Equation 25
GREEK
T
We can take
Equation 13
I&EC FUNDAMENTALS
(1) Bohr, N., Phtl. Trans. Roy. SOC.(London) A209, 281 (1909). (2) Gaskins. F. H.. Philippoff, W., Trans. SOC.Rheol. 3, 181 (1959). (3) Gill. S. J., Gavis. J.. J . Polymer Sct. 20, 287 (1956). (4) Hansen, R. S., Purchase, M. E.? Wallace, T. C., Woody, R . I\'., J . Phys. Chem. 62,210 (1958). (5) Metzner, A. B.: Houghton, LV. T., Sailor, R. A . , White, J. L.: Trans. SOL.Rheol. 5 , 133 (1961). (6).Middleman, S., D. Eng. dissertation, Johns Hopkins University: Baltimore, 1961.
RECEIVED for review March 25, 1963 ACCEPTED September 30. 1963 \Volk supported by a grant from the National Science Foundation.
