Ind. Eng. Chem. Process
Des. Dev. 1981, 20,391-397
391
Gas-Non-Newtonian Liquid Two-Phase Flow in Helical Coils E. A. Mujawar and M. Raja Rao’ Depam“
of Chemical Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India
The Lockhart-Martinelli type of approach has been extended for the flow of a gas-nowNewtonian liquid (power law type) twcbphase flow system in helical coils. Experimental work on flow pattern, holdup, and isothermal pressure drop at 30 OC was carried out in four helical coils having coil curvature ratios of 0.0100, 0.0198, 0.0476, and 0.0695 with air-water and air with each of the aqueous polymer solutions of 0.3 and 0.5 % (w/w) of sodium alginate and 0.5 and 1.O% of sodium carboxymethyl cellulose as test liquids. Suitable design correlations are proposed to predict the holdup and two-phase pressure drop in helical coils.
Literature Review The published information on gas-non-Newt” liquid two-phase flow is very scanty and hence the present work was undertaken to modify the Lockhart-Martinelli type of analysis for gas-non-Newtonian liquid (power law type) two-phase flow in helical coils and to obtain design correlations for holdup and frictional pressure drop, based on extensive experimental results. Flow Patterns. Rippel et al. (1966) conducted experiments for the downward flow of air-water, helium-water, Freon-12-water and air-2-propanol mixtures in a single coil of curvature ratio of 0.0625. The authors did not provide any calming section. The flow patterns observed were bubble, slug, stratified, wavy, semiannular, and annular. Banerjee et al. (1967) noticed an interesting phenomenon of film inversion, while conducting experiments in three helical coils of curvature ratio, 0.052,0.0695, and 0.104 with the air-water system. It was observed that for certain ranges of liquid and gas flow rates, the liquid travelled on the inner wall of the tube. This phenomenon was called “film inversion”. Banerjee et al. (1969) subsequently reported their findings on 13 different coils of curvature ratios ranging from 0.0917 to 0.275 with air-water and air-oil mixtures and concluded that the conventional flow patterns found in straight horizontal tubes were all found to occur in helical coils. Akagawa et al. (1972) made very useful studies on airwater flow in two helical coils of curvature ratios of (1:ll) and (1:22.7). These authors reported for the first time that the two-phase flow will be stabilized after a coiled tube length to diameter ratio greater than 90. The flow patterns observed were similar to those in straight horizontal tubes. Dayasagar (1975) carried out experimental work on four helical coils with air-water, a i d 1% glycerol, and air-89% glycerol systems and reported three main flow patterns, namely plug, slug, and annular. Ramaniah and Satyanarayan (1976) reported their limited studies with air-CMC mixtures in four helical coils having curvature ratios between 0.0505 and 0.0929. The variation in the coil curvature ratio was thus less than twofold. Further, these authors had used only one type of non-Newtonian liquid and its concentration is not specified, though different concentrations of carboxymethyl cellulose (CMC) were used. From the schematic diagram of the report, it is clear that they have not provided any entrance length to stabilize the flow pattern. Nonetheless, the authors had observed three types (slug, wavy, and annular) of flow patterns. Reddy and Satyanarayan (1977) have recently reported their studies on air-water systems using six different coils 0196-4305/81/1120-0391$01.25/0
in the coil curvature ratio range between 0.048 and 0.094. The flow patterns observed were slug and annular. Thus, it is seen from the literature that the flow patterns observed in helical coils were essentially similar to those observed in straight horizontal tubes. The flow in a helical coil involves a combination of horizontal and vertical tube flow but as horizontal distance travelled per turn is much greater than the vertical distance, the flow patterns in helical coils closely resemble those in straight horizontal tubes. Holdup. Rippel et al. (1966) found that the in situ liquid volume fraction EL, measured experimentally in a helical coil, was in good agreement with that predicted by the Lockhart-Martinelli (1949) curve for-straight horizontal tubes, a t higher valves of X,but EL values were lower at lower values of X. The authors attributed this difference to the downward flow orientation in the helical coil. Further, it was pointed out that the liquid holdup was influenced by liquid properties rather than gas properties. Banerjee et al. (1969) reported that their experimental values of EL agreed with those predicted by the Lockhart-Martinelli curve within *30%. Akagawa et al. (1972) found that the mean volume fraction of gas, EG, could be correlated by the equation EG = 0.82C~ (1) which was proposed by Hughmark (1962) for a straight horizontal tube. Reddy and Satyanarayan (1977) proposed an equation similar to eq 1except that they changed the constant from 0.82 to 0.778. These authors found that their experimental values of ELwere lower than those predicted by the Lockhart-Martinelli curve. It is evident from the literature that no work has been reported so far on holdup in helical coils for gas-non-Newtonian liquid systems. Pressure Drop. Rippel et al. (1966) and Owhadi et al. (1968) found satisfactory agreement between their resulta of two-phase flow pressure drop in a helical coil and the Lockhart-Martinelli correlation with a modified definition of the parameter X,for helical coils as X, = [ ( ~ ~ / A J W L , / ( ~ ~ / W S G , I (2) Rippel et al. (1966) introduced the drag coefficient approach for correlating two-phase flow pressure drop. According to these authors, the pressure drop in two-phase flow can be thought of as the friction loss in the dry gas phase plus an additional pressure loss when the liquid is introduced
0 1981 American Chemical Society
392
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
(fd"d) obtained from experiments from eq 3 was correlated with liquid input volume fraction CL. The correlations proposed by these authors are: for annular flow
coiled tubes for flow of non-Newtonian fluids of power law type is given by Mujawar and Rao (1978) fc
-
= $ (n?(De?+("')
fs
(10)
i.e. for bubble and slug flow
For coiled tubes for stratified flow
The success of these three equations depends upon the reproducibility of the specific flow pattern. Banerjee et al. (1969) reported that the experimental two-phase flow pressure drop results on liquid-gas being in a turbulent-turbulent flow regime in helical coils could be predicted by Lockhart-Martinelli's turbulent-turbulent flow regime curve, whereas results of the viscous-turbulent flow regime were satisfactorily correlated by the viscousviscous regime curve of Lockhart-Martinelli. The authors have not indicated what criterion is to be used for laminar and turubulent flow in helical coils while two phases are flowing. Akagawa et al. (1972) proposed a new correlation for two-phase pressure drop as
The success of the eq 7 depends upon the accuracy with which one can predict (AP)Tp,, as there is hardly any generalized equation to predict it within the accuracy of &5%. Further, the form of the eq 7 is such as to involve tedious calculations. Ramaniah and Satyanarayan (1976) proposed the equation
4Lc = 0.732X,4,505(d/D)4.5
(8)
I t is not specified for which flow regime this eq 8 is applicable. Both 4Lcand X , contain the functional term ( d / D ) and hence it does not look very logical to introduce another functional term like (d/D)4.5 separately in eq 8. Reddy and Satyanarayan (1977) reported a correlation similar to eq 8 for turbulent-turbulent flow regime
bLC= 4.02X;0.375
(9)
What criterion was applied to know whether the given phase is in the laminar or turbulent flow condition was not reported by these authors. Thus, it is clear from the critical review of the available literature that Lockhart-Martinelli correlations modified for coiled tubes seem to be applicable to predict two-phase pressure drop of gas-Newtonian liquid systems and no attempt was made to test them for gas-non-Newtonian liquid systems. Theoretical Considerations The basic assumptions of Lockhart and Martinelli (1949) are used in the present analysis for gas-non-Newtonian liquid two-phase flow in coiled tubes for four cases of flow regime. Case (i). Liquid Viscous-Gas Viscous i.e., MsL < 1000; MSG < 1000. The laminar flow friction factor in
Substituting for fc and VL in eq 12 and then rearranging gives
Similarly (14)
-
2 X 16$ytW~P
gc(D)0.5'(7r/4)PpLW
(15)
Multiplying both numerator and denominator of the right-hand side of eq 13 by d' and then substituting eq 15, we get
Similarly
Ind. Eng. Chem. Process Des. Dev., Vol. 20,No. 2, 1981
394
F r
b
c
~
03
o
1
I
d
10'
VSG*1O
L
- 1 C '/.S:WnC ______
AIR
,
I
3 33r
132
13
cm1s
SY:TEH
133 V5'.
Figure 1. Holdup and pressure drop data in helical coil of ( d / D ) = 0.0695 for air-water system.
gas and liquid flow rates gets disturbed by the action of the centrifugal force in helical coils,resulting in early onset of slug flow. Thus the slug flow was found to be more dominant in the helical coils and the stratified flow was confined to a narrow gas rate range of 150 to 400 cm/s, depending upon the coil curvature ratio. In annular flow observed in straight tubes, the gas flows through the central core and the liquid flows as a film on the periphery of the tube. However, due to secondary flow effect in a helical coil, some of the liquid from the film gets finely dispersed in the gas core, giving annular mist flow. The loci of the transitions of different flow pattems are marked in Figure 2 for a helical coil of the highest coil curvature ratio of 0.0695 used in the present investigation. A similar trend was observed with the other three helical coils, having curvature ratios of 0.0476,0.0198,and 0.010. From the observed flow pattems for air-non-Newtonian liquid mixtures, it could be seen that the nature and trends of flow pattern do not seem to have a pronounced effect. The transitions of the flow patterns are shown in Figure 2 for a representative gas-non-Newtonian liquid system in a helical coil of ( d / D ) = 0.0695. Holdup. This characteristic parameter is usually expressed in terms of holdup ratio, H defined as
This helps us to understand the holdup phenomena only in a qualitative way. Figures 1and 2 show the variation of H with VsLfor gas-Newtonian and gas-non-Newtonian liquid systems, respectively. Figure 1 reveals that the holdup ratio for an air-water system increases continuously with superficial velocity of air for any superficial velocity of water, but it starts decreasing in annular-mist flow. The maxima-minima observed in straight horizontal tubes in stratified, wavy flow patterns seemed not to be realized in a coiled tube. This might be due to the effect of curved flow geometry which helps to maintain plug flow conditions
13
cm
A I.4
i
Figure 2. Holdup and pressure drop data in helical coil of ( d / D ) = 0.0695 for air-1.0% SCMC system.
b---
LOCKHART MARTlMLLl CURVE
IO
PPI
Xc
Figure 3. Holdup resulta for flow of gas-Newtonian and gas-nonNewtonian liquid systems in helical coils.
rather than ripply and wavy interfaces in helical coils. The experimental results of in situ volume fraction of liquid, ELare plotted against X,for the four helical coils with all the test liquids in Figure 3. There is a satisfactory agreement between the experimental and predicted values of EL by the Lockhart-Martinelli correlation for gasNewtonian systems in all four helical coils, whereas EL values were found to be lower by 20 to 40% for gas-nonNewtonian liquid systems. This proves that the Lockhart-Martinelli parameter, X, alone is not sufficient to correlate ELfor the liquids with wide variation in viscosity or consistency index. When the complete data were carefully examined, it was found difficult to correlate E L
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981 395
Table IV. Correlated Values o f EL with XC for Gas-Non-Newtonian Liquid Systems
R E P R E S E N I S LOCKHART - M A R l l N E L L I CURVE R E P R E S E N I S P R E S E N T WORK
~~
0.094
0.12 1
Xc 0.7
0.175 0.26 0.33 2 5 10
0.41 0.53 0.62 20 50 100
v I S C O u s - V I s co u s
t
V I S C O U S -TURBULENT\
5’
39
,I
i
(A)
(b)
I
I
3.1
EG
,P
I to
0.1
10
1
EG
Figure 4. Variation of 4~~with in situ volume fraction of gas, for (a) v-t or t-t, (b) v-v or t-v flow regimes in coiled tubes.
with X and flow behavior index n’. However, a separate correlation for gas-non-Newtonian liquid systems has been proposed, in Figure 3, whose coordinates are given in Table IV. It was seen from eq 29 and Table I that there exists a functional relationship between and E@ An attempt was made in Figure 4 to determine this relationship. The results could be correlated by 1.85
4Gc
=
(EG)460
I
10
.
I
TURBULEN 1-TURBULENT
, IO
01
Figure 5. Comparison of pressure-drop results with LockhartMartinelli correlations for different flow regimes. Table V. Values o f C in Eq 31 for Different Flow Regimes
C
for viscous-viscous and turbulent-viscous flow regimes and 1.3
4GC
=
(E,)S.77
for viscous-turbulent and turbulent-turbulent flow regimes. Pressure Drop When two phases flow in a coiled tube, the effect of hydrostatic head must be taken into account. The exact treatment of the correction for hydrostatic head is shown by Mujawar (1978). Lockhart-Martinelli Approach. It was shown earlier in theoretical considerations that there exist a functional relationship between 4Lcand X,. To establish such a functional relationship, the experimental results are plotted in Figure 5 for a straight horizontal tube and helical coils with air-Newtonian liquid and air-non-Newtonian liquid systems. The experimental results show a definite relationship between 4Land X in a generalized way and give a satisfactory agreement with Lockhart-Martinelli curves, indicating thereby no effect of either flow geometry or nature of the test liquid, provided X is obtained properly by use of the most accurate correlations to predict single phase frictional pressure drop in the respective phases and provided a proper criterion of viscous and turubulent flow is used. Lockhart-Martinelli parameters are basically ratios, and the flow geometry, while affecting absolute level of pressure loss, does not alter the ratio of two-phase to single-phase pressure drop in a given flow configuration.
100
X
no. 1 2 3 4
flow regime of liquid and gas viscous-viscous viscous-turbulent turbulent-viscous turbulent-turbulent
present Lockhartwork Martinelli 6.6 12.0 8.0 12.7
5 12 10 20
The experimental result of the present work on twophase flow pressure drop was correlated by
where C is a constant for a particular flow regime, whose values are reported in Table V along with those obtained by Chisholm (1967) for the Lockhart-Martinelli curve for straight horizontal tubes. It is usually commented that the Lockhart-Martinelli approach does not take into account the effect of flow patterns. If one sees the four flow regimes of LockhartMartinelli very carefully, one can argue that these regimes are describing all the possible flow regimes. For example: (i) the first regime of liquid and gas being viscous-viscous covers mostly the elongated bubble; (ii) “viscousturbulent” covers the annular mist; (iii) “turbulentviscous” covers dispersed bubbles; and (iv) “turbulentturbulent” covers the slug flow patterns. Drag Coefficient Approach. Rippel et al. (1966) proposed this drag coefficient approach which should be applicable for any flow geometry. An attempt was made in Figure 6 to find out the relationship between Y and the liquid input volume fraction, CL, for all the test liquids in
396
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
SLUG
10
8
I’
ANNULAR-MIS:
FLOW
I ,
! 0 001
FLOW
, , , ,
,,I
,
, , , , ,,>I
01
0 01
CL:
,
1
1
1 / 1 1
I O
L l Q U l D INPUT VOLUME F R A C T ~ O N
Figure 6. Correlation of two-phase flow pressure-drop in annularmist flow by drag coefficient.
Table VI. The Values of C(n’)and b in Eq 32 no.
1 2 3
flow pattern annular mist slug stratified
C(n‘) 1.58(n’)-I1 5.66(n’)-’” 11.3(n’)-l4
b
0.886 1.25 1.53
all the test sections for annular-mist flow. The data points are satisfactorily fitted by essentially parallel straight lines, one for each test liquid, with average slope of 0.886. The intercepts represent the quantity C(n?pG/g,d. Since (pG/g,d) is constant in the present work, the quantity C(n? is thus determined. Similar treatment was followed for slug and stratified flow pattern results. It was seen that the flow geometry does not seem to have any significance, but the nature of the test liquid has a definite effect, if we treat the results by the approach of Rippel et al. (1966). An attempt was therefore made to determine the functional relationship between C(n? and n’ in Figure 7. The data points were fitted by straight lines, one for each flow pattern. Thus the correlation takes the following generalized form (32) The values of C(n? and b are tabulated in Table VI.
Conclusions (1)The nature and trends of the flow pattern in helical coils were essentially similar to those observed in straight horizontal tubes. (2) The liquid in situ volume fraction, ELfor gas-nonNewtonian liquid two-phase flow systems, was found to be consistently lower than that predicted by the Lockhart-Martinelli curve. Hence a new correlation for EL is proposed. (3) Equations 29 and 30 can be used to predict the in situ volume fraction of gas, EG.
(4) Two-phase flow pressure drop resulta were successfully correlated by eq 31, applicable for straight horizontal tubes and helical coils with Newtonian and non-Newtonian liquids. ( 5 ) Using gas phase drag coefficient approach for flow of gas-Newtonian and gas-non-Newtonian liquid mixtures, in straight horizontal tubes and helical coils, a correlation for two-phase pressure gradient represented by eq 32 was proposed.
Nomenclature A = cross-sectional area b = constant in eq 32 C = constant in eq 31 CG, CL = input volume fraction of gas and liquid D = helix diameter of the coiled tube d = inside diameter of the tube dG, dL = equivalent diameter for the tube cross section oc-
cupied by gas and liquid phase, respectively
E = cross-sectionalaverage in situ volume fraction of a phase, averaged over a finite length of tube
f = Fanning’s friction factor f d = drag coefficient g, = dimensional conversion factor H = holdup ratio defined by eq 28 AL = length of the test section n, n’ = flow behavior index p = function of n’ [= 2 - 2t: n’t] Ai’ = pressure drop over length AL q = function of n’[= 1 - t: n%] r = function of n’ [= 5 - 4 t: +- 3n’l - 0.591 S = function of n’ [= 2 - 2r + n T ] t = function of n’ [= 1 + nT - r] u = function of n’ [= 5 + 3nT - 4r - x ] V = cross-sectional average velocity VG, VL = in situ velocity of gas and liquid VSG, VSL = superficial velocity of the gas and liquid
+ +
Ind. Eng. Chem. Process Des. Dev. 1981, 20, 397-399
W = mass flow rate
X = Lockhart-Martinelli parameter
V = viscous (=[(hp/hL)s~/(AP/
Dimensionless Groups ( d / D ) = coil curvature ratio De’ = modified Dean number = Re,,,(d/D)1/2 = (CP’V(~-~’)-
U ) S G l ‘I2
Y = function =
307
[ ( h p / h L ) T p - (hp/u)SG]/(vSG)2
Greek Letters
p ) / (gcK%(n’-1))
a, @ = flow configuration factors a*, @*= function of n’ in eq 26
M = dimensionless number proposed by Mujawar and Rao (1978) Re = Reynolds number (dVp)/p
I’ = function of n’ defined in equation, f, = s2(d/D)0.5/
[ R (d/D)”lr Ztzner-Reed consistency term [= g&’ 8(”’-l) 6 = function of n‘ [= 1 - $1 I.L = viscosity of fluid p = density $ = Lockhart-Martinelli parameter [ ( A P / A L ) T ~ / ( A P / u)sP11’2 = function of n’ defined in eq 10 Q,w = constants x = function of n’ (= 0.5 - wr) Subscripts C = coil G = gas gen = generalized L = liquid S = superficial, single-phase s = straight tube TP = two-phase t = turbulent
Literature Cited
y =
Akagawa, K.; Sakaguchi, T.; Veda, M. Bull. J S M . 1972, 74, 564. knerjee, S.; Rhodes, E.; Scott, D. S. AfChEJ. 1967, 13, 189. Banerjee, S.; Rhodes, E.; Scott, D. S. Can. J . Chem. Eng. 1969, 47, 445. Chlsholm, D. Trans. ASME 1967, 80, 276. Dayasagar, I. Ph.D. Thesis, Indian Instltute of Technology, Bombay, 1975. Hughmark, G. A. Chem. Eng. Prog. 1962, 58(4), 62. Ito, H. Trans. A S M , J. BasiCEng. 1959, 81, 123. Lockhart, R. W.; Martlnelii, R. C. Chem. Eng. frog. 1949. 45, 39. Mashelkar, R. A.; Devarajan, G. V. Trans. Insf. Chem. Eng. 1977, 55, 29. Mulawar, 8. A. Ph.D. Thesis, Indian Institute of Technology, Bombay, 1978. Mujawar, B. A.; Rao, M. R. Ind. Eng. Chem. Process D e s . D e v . 1978, 17, 22. Owhadi, A.; Beii, K. J.; Craln, B., Jr. Int. J. Heat Mass Transfer 1966, 1 7 , 1779. Ramaniah, P.; Satyanarayan, A. Chem. Pet. J. (In&) 1976, 7(6), 3. Reddy, C. J. M.; Satyanarayan, A. J. Insfn. Eng. (India) 1977, 58, Part CH I Rlppel, 0. R.; E&, C. M.; Jordon, H. 8. I&. Eng. Chem. Process Des. D e v . 1966, 5, 32.
+
Received for review June 3, 1980 Accepted December 8, 1980
COMMUNICATIONS Critical Catalyst Concentration in the Liqufd-Phase Oxidation of Hydrocarbons A method to relate the critical catalyst concentration (CCC)to the length of the induction period, 71,the concentration of the hydroperoxide of the hydrocarbon being oxidized in the liquid phase, and an adsorption type constant, A ,
is presented for the heterogeneously catalyzed liquid-phase oxidation of hydrocarbons involving alkylperoxy (ROO-) free-radical termination. CCC is inversely proportional to A and increases as temperature increases. The method is judiciously applied to liquid-phase cumene and cyclohexene oxidations.
Introduction The critical catalyst concentration (CCC) phenomenon has been observed in the heterogeneously catalyzed liquid-phase free-radical oxidation of hydrocarbons (Meyer et al., 1965; Mukherjee and Graydon, 1967; Bacherikova et al., 1971; Evmenenko et al., 1972; Neuberg and Graydon, 1972; Gorokhovatskii and Pyatnitskaya, 1972; Gorokhovatskii, 1973a,b; Varma and Graydon, 1973; Neuberg et al., 1975; Mikhalovskii et al., 1976). A sudden and significant (almost catastrophic) change in rate for a slight change in catalyst concentration (or other rate determining parameter) represents a critical phenomenon and is a characteristic of branched-chain reactions (Gorokhovatakii and Pyatnitskaya, 1972). It appears that the CCC phenomenon is related not only to either the oxidizable hydrocarbon or the catalyst taken alone but, more appropriately, to the system taken as a whole (Gorokhovatskii, 1973b). Apparently, the “dual function” of free-radical initiation and termination exhibited by the compounds of transition metals generally used as oxidation catalysts is also essential for CCC. The “dual function” prevalent 0196-4305/81/1120-0397$01.25/0
predominantly at high catalyst concentrations in cumene (Gorokhovatskii, 1973a) and phenol (Sadana, 1979) oxidations, increases the length of the induction period rI with increasing catalyst loading in heterogeneously catalyzed cumene (Evmenenko et al., 1972; Gorokhovatskii and Pyatnitskaya, 1972) and phenol (Sadana, 1979) oxidations and in homogeneously catalyzed n-decane oxidation (Knorre et al., 1959). Similar increases in rI with an increase in the “nonactive” catalyst support (tending to decrease the “free” hydroperoxide of the hydrocarbon being oxidized in the liquid phase) has been reported in the oxidation of tetralin (Mukherjee and Graydon, 19671, cumene (Vreugdenhil, 1973), and phenol (Sadana and Katzer, 1974a). Recently, it was shown that the CCC value obtained (Sadana, 1979) from the available T I data for aqueousphase phenol oxidation (Sadana and Katzer, 1974a) (wherein the phenoxy (R.,alkoxy) radical terminates on the catalyst surface) compared very favorably with the CCC value obtained from the expression developed for the kinetic chain length (when it equals 0.5 (Neuberg and 0
1981 American Chemical Society
