TWO-PHASE GAS-LIQUID CONVECTION HEAT TRANSFER A Correlation E. J .
D A V I S A N D
M . M . D A V I D '
Dejartment o j Chemical Engineering, Gonraga C'niwrsitj. Spokane. Nhsh
A slip model, previously developed, has been used to develop an empirical correlation for two-phase gasliquid heat transfer data. The data o f many maior investigators have been analyzed to determine and to extract those for which nucleate boiling d i d not occur. These experimental data have been compared with the results predicted from the proposed correlation. The data in the purely convective heat transfer region (no nucleate boiling) have been predicted within an average absolute error o f 6 to 17% over a wide range of tube sizes, flow rates, pressures, and heat fluxes, and for vapor mass fractions from about 0.10 to the d r y - w a l l or liquid-deficient condition. The data correlated b y the proposed equation generally correspond to annular or mist-annular flow conditions. The results for the steam-water system are in good agreement with those for the air-water system that have been published.
frtquent occurrence of tivo-phase gas-liquid flo\\- in T.industrial .. heat exchange equipment and recent interest in HE
thin film cooling have prompted a large number of investigations iii t\\o-phase heat transfer since the 1930's. T h e literature survey\ o f Collier (3) and David (6) include the \vork u p to 1957. but Shrock aiid Grossnian (30),Sani ( 2 9 ) >Kvamme (22): Groothui, and Ilendal (17).Davis and David ( 6 ) . Parker and CFrobh ( 2 8 ) . Bennett rt 01 ( 3 ) .Silvehtri et ul. (37),Dukler ( 1 2 ) , Chcn ( I ) . and Andcrbon, Haseldcn. and Xfantzouranis ( I ! 1) have niade notru.orthy contributions to the knoivledge of t\vophaw hrat transfer sincr that time. Early investigators in the field were concerned with hrat exchange- in rvaporators and retioilers and reported only average values or ove:--all heat transfei coefficieiits. T o complicate d n y critical andlysis of the data, sorne authors failed t u distinguish betiveen nucleate boiling and convection heat ti,anafrr) Ivhich both occurred in the equipment. I 7nless point hrat transfer coeflicit3rits are reported, the data are of little u b e in designing equipment to operate in a range other than thal studied bb- the investigator. 'l'he fiist author to present a conipreheiisive and complete experirnrntal inveatigarion of t\\o-phase heat transfer \vas Denglrr ( / U . I I). who measured point coefficients in a vertical jacketed tube under thr conditions summarized in Table I. He recognized the existence of a nucleate boiling zone and a convrction-controlled zone, and also found that a dry-\vall condition occurred at high vapor fractions. Introducing a nonvolatile radioactive tracer into the liquid stream, he determined that a liquid-deficieiit region occurred. as evidenced by a n increased count a t the wall due to deposition of the salt on the \Val1 He also applied a radioactive tracer technique to measure liquid holdup. Anderson. Haselden, and Mantmuranis ( 7 ) tabulated the results of many of the correlations that have been proposed for calculatiiig heat tramfer coefficients in long tube evaporators. Little agreement exists among the various correlations. l ' h q also developed a theory for iipivard, annular floiv of a gasliquid mixture in a vertical tube using the treatment of von Karnian in applying the analogy between momentum transfer and heat t i anafer. \-oil Karman's universal velocitl- profile \\'as assunird to dppl>- to the liquid film, and the authors I Prvsent address. Lkpartincnr of Chernical Enginerring, Unit r s i r y of \Vasliington, Srartlr 5, [Vash.
I
assumed that the eddy diffusivities for hggt transfer and monientuin transfer are equal. T h e authors attempted to substantiate their theory n i t h the data of Dengler (70) and Lee (23),but the data \\-ere too inaccurate to confirm the theory. Anderson, Haselden, and Mantzouranis (2) published the results of a more accurate experimental study in which they rneasured point heat transfcr coefficients for the evaporation of water in a vertical tube. Qualitatively the results \\-ere similar to Dengler's. The theory developed in the earlier paper did not predict film temperature drops accurately: so the authors proposed an empirical correlation to account for the discrepancy. Dengler's results were not correlated by the proposrd equation. Chen ( - I ) recently compared the correlations of Denglrr and Addorna ( 7 1 ) . Guerrieri and Talty (78), Bennett et al. ( 3 ) :and Shrock and Grossman (30) with the experimental results of those investigators and the data of Sani ( 2 9 ) . He found existing correlations to be unsatisfactory and proposed a correlation for the hrat transfer coefficient \\-hich is based upon macroconvective and microconvective effects. 'l'he macroconvective term is a modified Dittus-Boelter equation, and the microconvective term is based on the Forster-Zuber formulation for pool boiling ( 7 6 ) . T h e correlation, included in Table 11, predicts the data \\ith an average deviation of + l l ~ , . Hewitt ( 7 9 ) has analyzed cocurrent uplvard flow of twophase gas-liquid mixtures using Dukler's approach ( 7 2 ) . Using DeisJer's equation ( 9 ) for the shear stress in the liquid film near the \\.all and von Karman's expression for the shear stress farther from the \\-all. he developed a relationship for the liquid film thickness in terms of the von Karman dimensionless vrlocity and dimensionless distance parameters. The approach is extended to Aoiv with heat transfer similar to the lvork of Anderson and Dukler; hoivever. Hewitt did not compare results of his analysia with experimental data. T h e application of the von Karman universal velocity profile to the annular film is questionable. Dukler (73) first applied it in developing his theory of film thickness in thin film flow. For the case of negligible gas flow the film thickness was predicted accurately. However, the assumption has not been tested under conditions of considerable interfacial shear bet\veen the vapor core and the annular film. i\ further difficulty, mentioned by Anderson. is that of the very substantial ripples that occur on the liquid surface. Considering these difficult problems it is not unreasonable that Anderson's theor)- failed VOL.
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111
Table 1.
Variables Studied by Investigators of Two-Phase Convection Heat Transfer
P.S.I . A .
Anderson et al. Bennett et al.
System Steam-water Steam-water
20-1 20 15-35
Mass Flow Rate, Lb./Hr.-Sq. Ft. x 1032.8-656 51.5-217
Davis
Steam-water
25-1 50
50-600
0.30-0,90
50-260
Dengler and Lee Fikry Groothuis and Hendal Kvamme Mumm Parker and Grosh Silvestri et al.
Steam-water Steam-water Air-water
7-40 15-25 14.7
44-1010 41.4-74.7 112-621
0.07-0.90 0,024.82 0-0.20
0-200 6.0-30 Not reported
0.0833 0.0208 0,0460
Tube Orientation Vertical Vertical (annulus) Horizontal (rectangular duct) Vertical Horizontal Vertical
Steam-water Steam-water Steam-water Steam-water
18-87 45-200 49-61 15-75
52-182 252-1008 37-73 73.6-162 (steam) 51.5-332 (water)
0-1 , 00 0-0.60 0.89-0.99 0.25-1 .OO
9.4-50.2 50-250 3.0-20.7 0-1427
0.0208 0.0387 0.0833 0.0164
Horizontal Horizontal Vertical Vertical
Pressure,
to agree with the experimental findings. Because of the numerous assumptions required to predict the thickness of liquid films in two-phase vapor-liquid flow, and because of the complexities of the mathematical model for heat transfer to such flow, an alternative and empirical approach to the study of two-phase convection heat transfer is needed for design purposes. Such an approach is developed in this paper. The data of the major investigators are analyzed and correlated over a wide range of conditions by means of a previously developed model ( 8 ) . Heat Transfer Mechanisms and Regimes
Several investigators (2, 3, 7, 70, 7 7 , 75, 77, 78, 22, 23, 26-37) have measured point heat transfer coefficients in the study of heat transfer to vapor-liquid mixtures flowing in ducts. T h e systems and variables studied are listed in Table I, and the proposed correlations for heat transfer coefficients are listed in Table 11. Dengler’s data (70) are typical of the results obtained by the investigators. At low flow rates and low vapor fractions the heat transfer coefficient was found to depend upon the heat flux and the temperature difference between the wall and the bulk stream (assumed to be the saturation temperature in a single-component system), At higher vapor fractions and higher mass flow rates the heat transfer coefficient was found to be independent of the heat flux, which is similar to the situation found in single-phase flow. Dengler correctly interpreted the dependence of the coefficient on the heat flux to mean that nucleate boiling made some contributioii to the total heat transfer. At very high vapor fractions (60 to 90% vapor and above) a dry-wall condition occurred and the heat transfer coefficient decreased to a value of the magnitude of a single gas-phase coefficient. Bennett’s findings ( 3 ) are in agreement with Dengler’s, except that the heat transfer coefficient was found to depend, to a small extent, on the heat flux in the convective region. In an earlier paper ( 8 ) the present authors published the results of an investigation of heat transfer to high quality steam-water mixtures flowing in an electrically heated, horizontal duct. Their results were similar to Dengler’s in that a nucleate boiling region, a purely convection-controlled region, and a liquid-deficient condition occurred as the steam quality was increased from 30% to 90y0. The point a t which nucleate boiling was suppressed and the quality at which the liquid112
I&EC FUNDAMENTALS
Vapor Mass Fraction 0.01-0.60 0-0.546
Heat Flux, B.t.u./Hr.-Sq. Ft.
x Io-
3.7-97 .O 63-1 58
Tube Diameter, Ft. 0.0387 0.0203 (equiv. diam.) 0.0324 (equiv. diam.)
deficient condition was reached depended upon the heat flux. Visual observation of the flow pattern indicated that annular and annular-mist flow occurred u p to the point at which the heat transfer coefficient reached a maximum, began to fluctuate widely, then dropped sharply. Little liquid, if any, could be observed on the wall at that point. For most of the runs of the authors. the temperaturedifference between the wall and bulk fluid was maintained low enough to prevent nucleate boiling; but, at low vapor fractions an’d high heat fluxes a nucleate boiling contribution occurred. For each operating condition of fixed pressure, total flow rate, and quality, the heat flux was varied to determine the dependence of the heat transfer coefficient on the heat flux. For most of the data the heat transfer coefficient was independent of the heat flux. I t was concluded by analogy with single-phase flow that the coefficient is only a function of the fluid properties and the dynamics of the system, but not the heat flux. The heat transfer coefficient is defined as: h = - - -qlA - q/A t w - t,,t At
(1)
T h e data were tested and smoothed by plotting heat flux against the temperature difference on a log-log plot for a fixed pressure, quality, and total flow rate. In the convection region the slope of the line through the data points is unity, indicating that the heat transfer coefficient is independent of the film temperature difference-Le.,
hAt = q / A log h
+ log At = log q / A
(2)
McAdams (25) states and several investigators ( 5 )show that in nucleate boiling the heat flux is a function of the temperature difference:
4 = KAtn A
(3)
-
Then
log 4-- = log K A
+ ().
log At
(4)
Experimentally this was shown to be the case a t low qualities and high heat fluxes, for the slopes of the curves of log q / A us. log At were found to be greater than unity. Figure 1 shows a sample of the data obtained by Davis (7).
Table II.
Authors Bennett
Proposed Correlating Equations ( D z ) 0 . 8 1 (C$)0'4 hD _ - 0.06C".O);;( kL
et
k L
-----I
L
al.
Chen
hmic
= o'oo122
kLO .l Qcp LO . 4 6 p L 0 . 49gc0. 26 ~.SpL0.29~0.24pv0.Z4
( At)0.24X (AP)o.laS
where F and S are empirically correlated factors. (S = 0 when nucleate boiling is totally suppressed) Mass Fractim Steam
Dengler
:
0.50
Fikry 0.4 5 x 5 0.5 where C = 0.84 C = 0.887 0.55 5 x 5 0.65 C = 0.904 0.70 5 x 5 0.80
4 6 8 1 0 20 TEMPERATURE DIFFERENCE (T,-
Figure
Guerrieri and Talty
1.
Sample
data
for
40 T,
60
),OF
steam-water system
Groothuis and Hendal
(F)"." Kvamme where R, = steam void fraction calculated from Martinelli-Lockhart correlation Mumm
h_II. k
= [4.3
+ 5.0
x
10-4
($)1'64
x] x
where V L =~ specific volume increase upon vaporization Shrock and Grossman where Bo Silvrstri et al.
=
(i) (A)
GraDhical correlation of
'The suppression of nucleate boiling is a complex phenomenon that depends upon the surface characteristics of the heating element, the surface tension, other physical properties of the liquid, and the flow characteristics of the system. Hsu (20) offered a very commendable theoretical analysis of the factors involved and Chen (4)developed an empirical correlation to account for boiling suppression, but there is not yet a generally accepted method to predict the suppression of nucleate boiling accurately. Because of the difficulty in predicting the thickness of the thermal boundary layer used in Hsu's analysis, the simple method of plotting the data, discussed above, has been used to analyze the data of other investigators for this paper, where there is doubt about the region in which the d a t a were taken. T h e upper limit of the two-phase convection heat transfer is the liquid-deficient region beyond the point of the maximum
heat transfer coefficient. In this region the coefficients are a magnitude lower than those obtained in annular or annularmist flow, and this liquid-deficient condition can lead to burnout in electrically heated systems. The stripping of the film from the wall has been studied by van Rossum (33) and Warner and Reese (34). Knuth (27) developed a theoretical treatment, but it is not yet possible to predict accurately the conditions under which liquid deficiency occurs. Vanderwater (32) proposed a model for predicting burnout in vaporliquid flow assuming a droplet diffusion mechanism for droplet transfer from the core to the wall with a re-entrainment function accounting for dispersal of liquid from the film into the core. The utility of Vanderwater's treatment is limited by lack of information on the droplet transfer coefficient and entrainment function. Parker and Grosh (28) used a model similar to that of Vanderwater in an attempt to predict the onset of mist flow, but were unable to substantiate their theory because of lack of information on droplet diffusion. However, they were able to predict qualitatively the longitudinal tube wall temperature profile. In the analysis of data for this paper the simple criterion used to extract data in the convection-controlled annular or annular-mist flow region was to plot the data a t a fixed flow rate and pressure as the heat transfer coefficient us. the vapor mass fraction. Only data below the point of the maximum heat transfer coefficient were used. Model
T h e results of all of the investigators mentioned indicate that a convection heat transfer region exists in two-phase gas-liquid flow heat transfer. In an earlier paper the authors developed a correlating equation based upon the following assumptions : Nucleate boiling does not occur. Heat transfer occurs from the wall to an annular liquid film, The heat transfer coefficient can be predicted from an equation similar to the Dittus-Boelter equation for single-phase flow in which the properties of the liquid are used for Lhe physical properties. A slip model can be used to relate the average liquid velocity to the average vapor velocity-i.e., '401. 3
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1964
113
hD -
=
B
b ( DPLCL y1)'(? )
L
kL
(5)
Incorporating Equation 9 into Equation 8 and determining the constant by fitting the data, the following correlating equation was developed :
Defining the slip ratio as:
and writing the liquid mass velocity PLGL = GL in terms of the vapor mass velocity and the slip ratio, the following is obtained:
(7) Assuming that a t high vapor fractions the vapor occupies nearly all of the duct, the superficial gas mass velocity can be written as:
G, = xGt Then Equation 5 becomes :
(2) ol.l (=
B
DGtx p~ -)a PL
(8)
(?)*
(9)
L
PB
I n general the slip ratio should be a function of the vapor and liquid flow rates, the system geometry, the physical properties of the system, and the flow pattern. For the range of conditions studied the authors empirically determined the terms B / a a of Equation 9 to be a function of the liquid-vapor density ratio (7, 8 ) :
Analysis of Data and Correlation of Results
The data of the authors are plotted in Figure 2 for the range of variables listed in Table I. The data are predicted to within an average absolute error of 8% by Equation 11, irhich is about the average reproducibility and experimental error in the data. Only data in the convection heat transfer region are plotted. The points a t very high flow rates (high vapor mass velocities) are badly scattered because the experimental error was highest in this range where temperature differences as low as 3.6' F. occurred. I n many of Dengler's runs the temperature difference was small enough to exclude nucleate boiling. Data he reported to be in the convection-controlled region are compared with Equation 11 on Figure 3. T h e data are predicted to lvithin an average absolute error of l i % > which is approximately the reproducibility of the data. Points 1 and 2 correspond to runs in irhich the temperature difference was enough to cause one to suspect nucleate boiling, and point 3 is from a run a t a quality of 0.707, which might be beyond the point of the maximum heat transfer coefficient. The correlation predicts results as well as that which Dengler proposed for the convection heat transfer region. Kvamme (22) studied heat traiisfer t o steam-water m i x t u r e a t high qualities as listed in Table I .
4000>i-r
Figure 2. 114
I&EC FUNDAMENTALS
-1I
Comparison of authors' data with Equation 1 1
200
400
600
1000
2000
Comparison of Dengler's data with Equation 1 1
Figure 3.
By maintaining small temperature differences he suppressed nucleate boiling in most of his runs. Kvamme includes the data of Yamazaki, who worked primarily in the low quality, nucleate boiling area. Some of Kvamme's data were badly scattered (as high as &40% in a few runs) either because of fluctuations in the wall temperature or error in the measurement of the smaller temperature differences, but the data have been smoothed by calculating the average heat transfer coefficient over his short (9.23-inch) test section ; however, the smoothed data can be considered to be point coefficients because the steam quality and pressure changes across the test section were small. Kvamme attempted to correlate his data by means of a void fraction model using the Lockhart-Martinelli correlation (24) for void fractions, but the scatter in the data preventrd adequate verification of the correlation. The smoothed data are plotted on Figure 4, and are correlated by Equation 11 within an average absolute error of 12%. Groothuis and Hendal (77) published a significant paper on heat transfer to air-water and air-oil mixtures. Their contribution is a complete and consistent study in convectioncontrolled two-phase heat transfer. KO nucleate boiling occurred, for thr tube wall was maintaincd at a temperature below the boiling point of the liquid in all runs. Their results indicated that the air-oil system behaved analogously to laminar flow of single-phase systems, and the air-water system behaved analogously to turbulent flow of single-phase systems. Although they studied low vapor mass fractions, the data are similar to those of the present authors. Using a Seider-Tate type equation for the correlation of the air-water data they proposed the following correlation : hD ---
= 0.029 (Re2)O.g' Prl!a ( M b / !Jff )0.14
k, where
Re? = Re'L
t R r I Q=
P,ic,D
(12)
+ PQQ& PP
-- ---
PI.
400
4000
The air-oil data were correlated by an equation similar to those for laminar flow of single phases. In the Groothuis-Hendal correlation the viscosity ratio term \\as included because the investigators worked with large
Figure 4.
6008001ooo
2000 3000
Comparison of Kvomme's data with Equation 1 1
temperature differences between the wall and the bulk stream. Equations 11 and 12 predict values for the Nusselt group that are in good agreement at all but very low vapor mass fractions. Figure 5 compares Equations 11 and 12 for the steam-water system at a pressure of 50 p.s.i.a. for qualities from 10 to 80% for a tube diameter of 0.5 inch. Equation 12 predicts Nusselt numbers 10 to 15% lower than Equation 11 for the data of Davis, and in general Equation 12 predicts Nusselt numbers 10 to 25% lower than Equation 11 a t vapor fractions greater than 0.20. At very low vapor fractions the equations deviate considerably, for Equation 11 is not valid when the liquid fills a substantial portion of the duct. As the quality approaches zero, the Reynolds number in the GroothuisHendal correlation approaches that for a single phase, and the liquid Reynolds number predominates. At high vapor fractions the vapor Reynolds number predominates, and the equation has a form much like Equation 11. T h e principal difference in the results of Groothuis and Hendal and the authors is that the liquid-deficient condition occurred a t much lower vapor fractions in the air-water system than in the steam-water system. The data of Groothuis indicate that a dry-wall condition occurred a t vapor mass fractions below 0.20. Anderson et al. found convection heat transfer to occur in the upper portion of their vertical tube. Data from their paper in which the steam mass fraction exceeded 0.10 have been plotted on Figure 6 to compare with Equation 11. The data are those for the top position in the tube, corresponding to section 13 in their work. Except for run 16, which might have been near the dry-wall condition, the Nusselt numhers are predicted to within an average absolute error of 10% for all of the data, in the specified vapor mass fraction rang?. Bennett rt a / . ( 3 ) have reported data for heat transfer to steam-water mixtures flowing in annuli. T h e results are similar to those of other investigators except for a slight dependence of thr heat transfer coefficient on the heat flux even in the purely convective region. Liquid deficiency occurred at about 0.65 mass fraction steam in their vertical section. '1 able I1 lists their correlation for the convection-controlled region, which is a modified Guerrieri and Talty equation (78). 'The dependence of the heat transfer coefficient on the heat flux is in disagreement with the results of the authors, but such VOL.
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115
clight depcndence can probably be correlated by considering similar situation in single-phase flow. Bennett used the ~iituc-Boelterq u a t i o n for calculating h L , the heat transfer c:ocflici.-im1 based upon liquid flow alone. If the Seider-Tate e q ~ i a i i o r iLvere used, the factor (pb/pE)'I,14would be introduced t o a l l o ~ for . the large difference in viscosity between the bulk liquid teinprrature and the wall temperature. This factor could ac'co!lnt for the increase in heat transfer coefficient as the lieat f l w (and hence temperature difference) increases. 'l'hc iiinitrd data tabulated by Bennett have been compared with Equation l l ? aud the results are given in Table 111. The averagr~absolutc r1 ror for this sampling of data is 5.2%. 21
When the rather drastic assumptions involved i r i developing Equation 11 from thenaive model are considered. it is intt-reSting that such a wid? range of data is correlated. 'L'he empirical expression for the slip ratio (Equation 10) could not be expected to hold for such 'widely varying conditions. Other data bear out this consideration. The data of Fikry ( 7 5 ) and M u m m (26, 27) deviate greatly from the values predicted by Equation 11: but they d o fit an equation of the same form:
where M is a constant sptxific to the system gcornetry. Fikry investigated the steam-water system over a iiari'ow range of total mass flow rates listed in Table I. The temperature differences between \Val1 and bulk stream were very small for most of his runs (2' to 30" F,), so nucleate boiling did n o t occur. Figure 7 shows that Fikry's data are correlated very well (6% average absolute error). except for t ~ v opoints a t vapor mass fractions of about 0.20, by letting A = 0.024 in Equation 13. A few data in the convection-controlled region of tLvo-phase heat transfer can be extracted from Mumm's work. He measured heat transfer coefficients to the steam-watrr system a t very high total mass velocities. The data have been analyzed by preparing log-log plots of heat flux us. tmmperature difference to determir;e what data were i i i thtr convrction
Table 111.
Comparison of Bennett's Sample Data with Equation 1 1 Annulus
wc-40
t- I L l I -
0
0.2
-4
I
I
I
I
0.4 06 0.8 MASS FRACTION VAPOR, x
Figure 5 . Compurison of with Equation 1 1
I
I I .o
I
-
1
Groothuis-Hendal correlation
D, Do
= 0.623 inch = 0.866 inch Gt = 170,000 Ib./hr. sq. ft.
(?Lot, (hL 199
210 222 236 250 262 268
175 194 210 224 242 258 272
116
I&EC
FUNDAMENTALS
t.
Figure 7. Comparison Equation 1 3
60
of
00 100
Fikry's
I
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 of distance from the tube exit. From this solution, the distance required for the diameter to become almost constant may b e predicted. The theoretical results are 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 ~
