Ind. Eng. Chem. Res. 1991,30,2172-2180
2172
Ethylene, Parahydrogen, Nitrogen, Nitrogen Trifluoride and Oxygen. J. Phys. Chem. Ref. Data 1982,11, Suppl. 1.
Soave, G. Application of a cubic equation of state to vapor-liquid equilbria of system containing polar compounds. Znd. Eng. Chem. Symp. Ser. 1979,No. 56, 1.2/1-1.2/16.Walas, S . M.Phase Equilibria in Chemical Engineering; Butterworth Boston, 1985. Younglove, B. A. Thermophysical Properties of Fluids. I. Argon,
Received for review March 20, 1990 Revised manuscript received October 17, 1990 Accepted April 22,1991
Heat Transfer to Air-Water Plug-Slug Flow in Horizontal Pipes Sanjeev D. Deshpande,* Alfred A. Bishop, and Bhalchandra M. Karandikar Chemical and Petroleum Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
Experimental heat-transfer data for air-water, two-phase flow in electrically heated, horizontal pipes were obtained. Stainless steel pipes of 0.028- and 0.057-m inside diameter were employed, and the plug/slug flow regime was studied. On the basis of the mixture velocity, the liquid was in turbulent flow and the gas was in transitional or turbulent flow. The water velocity range was 0.2-1.11 m/s, and the air velocities ranged up to 3.6 m/s. Time-averaged heat-transfer coefficients at the pipe top, bottom, and sides were measured at many axial positions. In two-phase flow situation, as much as a 50 K difference between pipe top and pipe bottom outside wall temperatures was recorded. Heat-transfer coefficients a t the pipe top ( h m )were lower than those at the pipe bottom (h?!). Both hTPTand hTPBincreased with increasing liquid velocity, and hm was relatively insensitive to changes in gas velocity. Slug flow hydrodynamic parameters such as frequency and liquid slug length to total slug length ( L s / h )ratio are important in understanding the heat-transfer phenomena. Therefore, heat-transfer correlations developed for pipe bottom, pipe top, and the overall average Nusselt numbers include the effects of these slug flow parameters. pande (1985);Bishop and Deshpande (1986)). During this investigation, circumferential variations in tube wall temperatures and heat-transfer coefficients in 0.028- and 0.057-m electrically heated horizontal pipes were observed. The results are discussed with reference to hydrodynamics and separate correlations for predicting heat-transfer coefficients at the top and bottom of the pipe are also presented. Hydrodynamics of the plug-slug flow is quite complex, and several flow parameters such as slug frequency ( v ) , ratio of liquid slug length to total length of the slug unit (Ls/LT),average liquid slug holdup (Rs), and liquid film velocity are considered important in characterizing the flow behavior. All these parameters, to a large extent, are dependent on the liquid and gas superficial velocities, pipe diameter, and fluid properties. Experimental and theoretical investigations to improve understanding and predictability of these parameters are in progress (Taitel and Dukler (1977); Barnea and Brauner (1985); Tronconi ( 1990)). Experimental studies for Newtonian systems (Gregory and Scott (1969); Heywood and Richardson (1979)) have concluded that at a constant liquid flow rate the slug frequency initially decreases as the gas velocity increases and after reaching a certain minimum value, frequency increases again. The slug frequency increases with increasing liquid velocity and is lower for a larger pipe diameter at the comparable conditions of liquid and gas velocities. Heywood and Richardson (1979) have demonstrated that over a large range of VSL/VNs,the ratio of liquid slug length to total length of the slug unit is a linear function of VsL/VNs ratio. Because of the analogous nature of momentum and heat transfer, it is obvious that the heat-transfer characteristics depend on flow parameters to a considerable degree and, therefore, they should be included in any correlation for plug4ug flow heat transfer. Several researchers have reported analytical and/or experimental studies of heat transfer to horizontal, gasliquid plug-slug flow in the past (Oliver and Wright (1964); Lunde (1961); Johnson and Abou-Sabe (1952); Fried
Introduction and Background Gas-liquid two-phase flow in pipes is frequently encountered in chemical, food, and pharmaceutical industries. Nonuniform heat transfer would affect the reaction rates and the product quality, and it may also cause local charring in processes such as coal liquefaction. Although problems concerned with heat transfer to such systems are important, two-phase, two-component heat transfer without boiling has received considerably less attention compared to single-component heat transfer involving a phase change. Plug-slug flow that can exist over a wide range of phase velocities is characterized by the alternate passage of liquid slugs and gas bubbles or slugs. The liquid slug that may be aerated at high gas velocities is considered to fill the entire cross section of the pipe, whereas a gas slug occupies only a fraction of the cross section. Because of the effect of gravity, gas slugs are concentrated in the upper half of the pipe in horizontal flow. Thus, gas and liquid slugs are alternately in contact with the pipe top, and a continuous liquid phase is in contact with the pipe bottom. The liquid slug picks up liquid at the slug nose and sheds back the liquid in the form of slow-moving film (Hubbard and Dukler (1975)). (For viscous liquids, there might be a thin, relatively stationary liquid film at the pipe top.) Thus, the unsteady nature of slug flow may cause substantial variations in wall temperatures and in peripheral heattransfer rates, especially when a constant heat flux condition is employed. In this paper, time-averaged heat-transfer characteristics of air-water, plug-slug (or intermittent) flow under a constant heat flux condition are reported. These measurementa were made to provide a sound comparative basis for the heat-transfer testa involving non-Newtonian liquid-air mixtures flowing through horizontal pipes (Desh-
* To whom correspondence should be addressed at Hoechst Celanese Corporation, P.O.Box 9077,Corpus Christi, TX 78469. 0888-5886/91/2630-2172$02.50/0
Q
1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 9,1991 2113 Table I. Summary of Plug-Slug Flow Heat-Transfer Studies in Horizontal Pipes (Airwater) pipe diam, type of heated investigator m heating length, m comments Constant Wall Temperature Steam 4.72 wall temperature variation noted Johnson-About-Sabe (1952) 0.022 Fried (1954) 0.019 Steam 4.72 wall temperature variation noted 1.31 air-glycerol system also studied 0.0064 water jacket Oliver-Wright (1964) Fedotkin-Zarudnev (1970) 0.01,0.022 water jacket 2.5 vertical flow also studied; several viscosity liquids used no exptl data reported data from previous studies correlated based on a physical model Lunde (1961) data from previous studies correlated by using in situ liquid Hughmark (1965) no exptl data reported velocity Shoham et al. (1982) Mehta et al. (1984)
0.0381 0.m
electrical heating blocks
Kago et al. (1986) Pletcher-McManus (1968)
0.0515 0.025
electrical electrical
Constant Heat Flux 1.76 time and space variation of heat transfer noted 1.83 very few measurements reported; circumferential tamp variation noted varying viscosity liquids studied 0.49 annular flow studied; circumferential temp variation not& 1.52 correlation presented for local and average coefficients
(1954);Hughmark (1965);Fedotkin and Zarudnev (1970); refer to Table I). However, mostly heat transfer at constant wall temperature was studied, and only the overall average (time and spatial average) heat-transfer coefficients were reported. Fried (1954) and Johnson and Abou-Sabe (1952)utilized several tube wall thermocouples and reported observing variations in wall temperature even though condensing steam was used as a heating medium. Pletcher and McManus (1968)reported existence of sizeable circumferential variations in the tube wall temperatures at low air flow rates during their experimental study of heat transfer to air-water annular flow. More recently, Shoham et al. (1982)observed fluctuations in wall temperature and in heat-transfer rate with time and around the pipe circumference in their extensive study. A few measurements were reported by Mehta (1984) for a 0.171-mdiameter pipe. Kago et al. (1986)observed differences in wall temperatures between the top and the bottom of the pipe during their experimental investigation of heat transfer to gas-liquid and gas-slurry systems. However, since these differences were relatively smaller (less than 7%) than the temperature difference between the heater wall and bulk liquid, they also presented the results in the form of average heat-transfer coefficients. It is clear that only recently have investigators started studying the circumferential variation in heat-transfer rates during plug-slug flow. Moreover, there are no correlations available to predict the heat-transfer Coefficients at various peripheral locations. It is extremely important to predict the variation in heat-transfer rates for good product quality in food and pharmaceutical industry and for the adequate mechanical design of tubes. This need is addressed by this investigation and correlations for pipe bottom, pipe top, and overall average heat-transfer coefficients are presented.
Experimental Apparatus and Procedure The experimental loop constructed is shown in Figure 1. The control panel and data recorder used to indicate and record all of the measured parameters are not included in Figure 1. [A detailed description of the apparatus and procedure is also available elsewhere (Deshpande (1985); Deshpande and Bishop (1987)).] Filtered distilled water was stored in a main holding tank and was circulated through the loop by a Moyno pump. Water flow through the test section was controlled by adjusting the flow through a bypass line. The volumetric flow rate of water was measured by an electromagnetic flow meter. Air was filtered and then regulated. The air flow was metered by two rotameters in parallel, and air was added to the flowing water after the liquid flow measurement. The air-water mixture after passing through the electrically heated stainless steel test section
A I
A
i
SIMPLIFIED HEAT TRANSFER L ~ O P SCHEMATIC DIAGRAM
r Disenaaaement +Surge f i n k
80 kW Power suppiy+
+D.c. Mixer
Pressure Gauge
Exchanger
-
Thermocouple Stations (axialtperipheral)
"
Transparent Test Section Holdup Tank (A)
Prehester
Mixing Tank (e) (with submerged pump)
8 MOYNO Pump Flow Meter (multiple range)
Figure 1. Schematic diagram of the experimental flow loop.
entered the disengagement tank. An air-disengagement or surge tank provided the air-free liquid and permitted a relatively pressure pulsation free operation. Water from the surge tank was cooled in the heat exchanger and then returned to the main holding tank. Thus, a recirculating mode of water was used during these tests. The electrically heated test sections consisted of two thin-walled 304 stainless steel pipes of 0.028-and 0.057-in inside diameter (wall thicknesses were 2.7 and 1.85 mm, respectively). The heated length was 1.7m. For the 0.028and 0.057-mpipes, LID values provided before the heattransfer section were 53 and 28, respectively. Several thermocouples were placed at seven axial locations having 0", 90", 180°,and/or 270° radial witions (shown in Figwe 2)to measure the outside tube wall temperature profiles. Thermocouple beads were made of iron-constantan (glass on glass, 30 gage) lead wire and were insulated from the test section by using a very thin mica sheet. Platinum resistance temperature detectors were used to measure the bulk fluid temperature at inlet and outlet of the heated test section. These detectors were calikrated by using a Leeds and Northrup standard NBS resiator to measure the water bath temperature. A dc rectifier was connected to copper bus bars attached to the test Eacticn through cables and thus provided the energy for heat transfer. Thus, a constant-heat-flux condition was achieved. The average heat flux was calculated from the current and voltage measurements and was checked by calculating the sensible heat transferred to the fluid mixture.
2174 Ind. Eng. Chem. Res., Vol. 30,No. 9,1991 28
23
25
21
0 24e22
2o919 18
Symbol Dia (m) 0 0 028 v 0 057 0028 v 0057
I&* 11
-
lead I
I
I
! n! I .
FLOW
h
I3
Correlation Dittus-Boeller Dittur-Boelter Petukhov Petukhov
0 0 0
0.028 m diameter pipe
0.8 ' ' 1 ' ' ' ' ' ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' ' ' 15,000 20.000 -25,000 30,000 35,OOC
Re
Figure 3. Comparison of experimental, single-phase heat-transfer coefficients with Dittue-Boelter and Petukhov correlation.
0.057 m diameter pipe Figure 2. Thermocouple arrangements for the test sections. All dimensions are in meters.
Table 11. Range of Parameters V, heat flux pipe ResL Res VSL, X lo4 X lo-' i.d.,m m/s m/s X lo4, W/m2 0.028 0.31-1.11 0.0-3.62 34-180 0.86-2.86 04.92 0.057 0.2-0.5 0.0-1.25 13-42 1.39-3.44 0-0.63
For each water flow rate, several tests were conducted by varying the gas velocity. In general, water was in turbulent flow, and the gas flow regime ranged from laminar to turbulent on the basis of the air superficial velocity. After the start of each run, sufficient time was allowed to attain a steady state, and then all of the parameters were recorded. Heat flux was adjusted so that the bulk fluid temperature difference between inlet and outlet of the test section was as large as possible. This ensured more accurate heat balances. Energy balances (after accounting for the heat loss through the insulation) in worst situations agreed within i 7 9 i (average absolute error was 3%). Errors in heat balances resulted primarily because of small fluctuations in water flow rate around its nominal value. The outside tube wall surface temperature was kept below 100 O C in order to avoid additional complications due to local boiling. Because of this constraint on heat fluxes that could be employed, measurements at higher liquid and gas velocities could not be made (refer to Table 11). During each run, time-invariant values of outside tube wall surface temperatures, bulk fluid temperature a t the inlet and outlet, and the electrical power supplied were measured. Pressure drop across the test section was measured by using manometers. The water and air flow meters were calibrated in the laboratory, and consistent agreement of the standard friction factor versus Reynolds number relationship for turbulent flow of water and air helped to confirm the flow measurement accuracy. Single-phase heat-transfer runs for water were conducted prior to these air-water tests to establish the measurement and recording procedures. Results and Discussion Measurement of tube wall temperatures during adiabatic runs conducted periodically for water alone showed no circumferential variations. This established the consistency of thermocouple readings. For heat-transfer runs of single-phase water the maximum difference between pipe bottom and pipe top temperatures was 2.8 "C for both pipe
sizes studied. During two-phase runs, the difference between pipe top and pipe bottom temperatures progressively increased as gas flow rate was increased for a given liquid flow rate. Differences as high as 50 K were recorded at higher gas velocities. Prior to the air-water heat-transfer tests, heat transfer to turbulent, single-phase flow of water was studied. The heat-transfer coefficients were compared with the widely used Dittus-Boelter correlation (McAdams (1954)). As shown in Figure 3, where the ratio of experimental to predicted heat-transfer coefficient is plotted against the Reynolds number, all of the values agree well within the accepted range of *30% of the Dittus-Boelter correlation. However, the experimental values are consistently higher. Similar trends have been reported by Bishop (1974) and by Allen and Eckert (1964). When compared with the predictions of the Petukhov (1970) correlation, data are in excellent agreement and are spread evenly on both sides of h,,/h,, = 1 line. Air-water heat-transfer tests were carried out in 0.028and 0.057-m-diameter pipes, and the range of parameters studied is given in Table 11. Heat-transfer coefficients at different circumferential positions were calculated from the measurements as outlined in the Appendix. The variation in heat-transfer coefficients in the axial direction was not significant, indicating that the two-phase flow was fully developed. Nevertheless, arithmetic average values of various axial locations at pipe bottom, pipe top, and sides of the pipe were calculated and are reported here. The axial average heat-transfer coefficients a t the top and bottom of the pipe for 0.028- and 0.057-m diameter pipes are shown respectively in Figures 4 and 5. It can be seen that two-phase heat-transfer coefficient at pipe top (hm) is always less and the two-phase heat-transfer coefficient at pipe bottom (hTPB)is higher than the single-phase heat-transfer coefficient value. The hTpBinitially increases more rapidly with the superficial gas velocity (Vsc),and then the rate of increase is smaller at higher V m The overall increase in hB can be attributed to an increase in mixture velocity as the gas velocity increases. For example, at VsL= 0.5 m/s; for V , = 0,hTpBis 2.8 W/m2 OC and for VsG= 0.5 m/s, hTPBis 4.2 W/m2 "C. This gives us the ratio of heat-transfer coefficients as h2/hl = 1.5. Assuming that h Vma8,h2/hl would be equal to 1.74. This compares very well with the actual value of 1.5. A slightly lower actual value than the predicted one indicates that the liquid film at the pipe bottom might be moving more slowly than the body of the slug. A reduction in the rate of increase of hTPBat higher V ~ could G be due to the fact that small gas bubbles get trapped in the liquid slug and the average liquid slug holdup (RS)is less than 1. Q:
Ind. Eng. Chem. Res., Vol. 30, No. 9,1991 2175 D:0.057 m, Vs~:0.3mls hsp:1671 Wlm2OC
0 Top Heat Transfer Coefficients
0 Bottom Heat Transfer Coefficients
D:0.028 m, Vs~:O.dlmls hsp:1611 Wlm2OC
Air-Water Data
0 2339 (2514)
L
-
vsG:o.39
r
~1935:
2339 (2164)
hTp:1864
(1911)
(1727)
2638
Superficial Gas Velocity, VSG
2656
U-
(m/s)
h~p:2028
Figure 4. Average axial heat-transfer coefficients at pipe top and pipe bottom as a function of gaa velocity (D= 0.028 m).
1529 (1624)
0 l o p Heat lransbr Coefficients 0 Bottom Heat Transfer Coefficients
vSG:o.95
1529 (1433)
4058
Air-Water Date
Figure 6. Effect of pipe diameter on heat-transfer coefficients at comparable liquid and gas velocities. All heat-transfer coefficients are in W/m2 "C. Table 111. Comparison of Measured b-
and bmR
hWB
b
X X lW3, Re& VsL, V,, W/mZoC m/s x ~ P W/m*"C @ m m/s At Comparable Mixture Reynolds Number 0.028 0.62 1.0 56.7 4.7 2.3 0.5 57.0 2.8 0.4 0.057 0.3
D,
0.028 0.057
At Comparable Liquid and Gas Velocities 3.7 1.0 0.3 1.0 45.5 0.3 1.0 92.6 3.6 0.3
@Re* = D V ~ L / ~V& L ; = VNS= VSL + Vm.
bB and hTpr are considerably smaller. 0 1
OO
0.5
I
I .o
I .4
Superficial Gas Velocity, VSG (m/s)
Figure 5. Average axial heat-transfer coefficients at pipe top and pipe bottom as a function of gas velocity (D= 0.057 m).
The hm values show a small decline with increasing There are two important factors that affect the heabtransfer coefficient at the top of the pipe: (a) increase in mixture velocity causes an increase in h m and (b) reduction in LS/LTand initial decrease in slug frequency (for a fmed liquid flow rate, as gas velocity is increased slug frequency initially decreases, reaches a minimum value, and then starts increasing again) would reduce the h T Therefore, due to these two compensating effects, hTPTis relatively insensitive to an increase in V D Both hTpr and h T p ~are higher for a higher liquid velocity ( VsL). Higher values of slug frequency for higher VsLmake hm larger, and the higher mixture velocity is responsible for higher
V,.
b B .
A comparison of hm and hmB can be made at a comparable mixture Reynolds number (refer to Table III). For the larger diameter pipe and lower liquid velocity, both
Both a larger pipe diameter and lower VsL cause a lower slug frequency, and this results in a lower value of hTm A large variation in bBand hTPTat a comparable mixture Reynolds number also indicates that the mixture Reynolds number alone cannot be the only governing correlating parameter for two-phase heat transfer. Moreover, flow parameters such as slug frequency and Ls/LT ratio should be included in any meaningful correlation. A comparison of h m and hpB is made at comparable liquid and gas superficial velocities for two pipe diameters as shown in Table 111. It is evident that the variation in hTpB is negligible, indicating a small effect of pipe diameter on bB. However, hm is much smaller for 0.057-m pipe as compared to 0.028-m pipe, reflecting a lower slug frequency and greater phase separation in larger pipe. A similar effect of pipe diameter is seen in Figure 6, where heat transfer coefficients at bottom, top, and sides of the pipe are shown for VsL = 0.3 m/s and at three comparable Vsc's. The measured heat-transfer coefficient values at the right and left sides of the pipe are written in parentheses below the arithmetic average valuea of theae two for each circular cross section. The right-hand and left-hand-side thermocouples were not exactly at the same
2176 Ind. Eng. Chem. Res., Vol. 30, No. 9,1991 Pipe
Diameter:0.057 m Vs~:0.5mls
o6lT
(3969) 3790
(3610)
3828
-()6:z33
(4005) 3780
bG”S
(3554)
Figure 7. Ratio of pipe bottom to pipe top heat-transfer coefficients versus the ratio of superficial gas-to-mixture velocities.
horizontal level. The left-hand-side thermocouple was slightly lower than the right-hand-side one. Therefore, it recorded a lower temperature reflecting a higher heattransfer coefficient value depending on the position of air-water interface. The hmB for 0.028- and 0.057-m-diameter pipes are almost equal. However, slightly lower values for the larger pipe show a small effect of pipe diameter consistent with a Dittus-Boelter type of relationship. There is a pronounced difference in hTpT values for the two pipe diameters. This is expected because of a lower frequency and partly due to greater segregation of phases in a larger pipe. Thus, for larger pipe diameters, a greater difference between h m and hmB is expected. The overall average heat-transfer coeffi_cients(arithmetic average of the circumferential values, hTp) are not a true measure of the local heat-transfer performance. Plots of the ratio of bottom-to-top heat-transfer coefficient, hTpB/hm, versus gas-to-mixture velocity ratio, V g / VNS, are drawn in Figure 7. The ratio V a l VNs equal to 0 corresponds to the single-phase liquid flow, and the ratio equal to 1corresponds to single-phase gas flow. In both limiting conditions, hm is expected to be equal to hmB provided free convection is not significant. For example, for a 0.057-m pipe during single-phase runs, experimentally measured hpB/hTPT ratios for 0.2 and 0.3 m/s water velocities were 0.99 and 0.986, respectively. The hmB is always greater than hm [This is in contrast to a situation where hTB < was found for the flow of viscous, non-Newtonian liquid and air flow (BishopDeshpande (1986)).] As VSL increases, the hTPB/hm ratio decreases, indicating a relatively higher increment in hm compared to that in bwThis result is primarily due to an increase in slug frequency with increasing VsL. At a fixed VSL, h n B / h m increases as VSG/VNSratio increases. This indicates a higher increment in hmB than in hm with increasing VSG. The higher mixture velocity increases the hwe, whereas two compensating factors (as discussed earlier) prevent a substantial improvement in h m As higher values of the VsG/VNs ratio are employed, higher gas flow would transform the slug flow into annular flow provided the liquid flow rate is high enough. However, even before annular flow is obtained, hTpr in slug flow is expected to increase more sharply as the slug frequency increases after the minimum in frequency is reached. In this investigation, high enough gas velocities were not used to see this increase in frequency and a corresponding increase in hTm The effect of VsL on h m and hTPBis shown in Figure 8. Here the circumferential heat-transfer coefficient values
4109
h~p:3138 (3898) 3492 0
-
3
4 (3085) 9 2
4593
(2070)
(1529) 3725
(3533)
(2595) 5150
Figure 8. Effect of liquid superficial velocity on bottom and top heat-transfercoefficients. All heabtransfer ccefficientaare in W/m2 OC.
are indicated around the circular cross sections for 0.057-m pipe for two different VsL’s as VSG increases. The hmB values consistently increase with increasing V, at a fixed Vsh For the VsL of 0.5 m/s, h m are higher than for VsL = 0.3 m/s. However, for VSL = 0.5 m/s, h m initially decreases owing to a decrease in frequency and then it increases, reflecting an increase in mixtFe velocity. The overall average heat-transfer coefficient, hTp increases with Vw at low values of V, and then & decreases at higher values of V., However, the variation in i;71.over the entire V, range is small, indicating a relative insensitivity toward vSG*
Correlation of Data It is clear that any correlation of two-phase, plug-slug flow data should include slug flow parameters such as slug frequency and/or the ratio of either liquid or gas slug length to the total length of the slug unit. Therefore, to incorporate these parameters in correlations presented here, experimental data and correlations for these parameters from the literature were utilized. On the basis of their extensive experimental work for the air-water system in 0.042-m-diameter pipe (VsL range 0.25-1.96 m/s and V, range 0.2-6 m/s), Heywood and Richardson (1979) showed that the ratio LS/LT varies linearly with VsL/ VN, ratio. The exact form of the correlation is Ls/LT = VSL/VNS - 0.1 for VsL/VNs > 0.1 (1) Heywood and Richardson (1979) used the mixture Froude number in correlating their slug frequency data. They
Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991 2177
10.2 0.4
0.6
0.8
(PrL)
0.4
1.0
VSL
(--o.I)
2.0 0.3
VNS
Figure 9. Correlation for pipe bottom heat-transfer coefficients.
Figure 10. Correlation for pipe top heat-transfer coefficients.
have developed a dimensional correlation for slug frequency under isothermal conditions: Y
= 0.0434[
(&)( vNS
8-
1.02
+ -5)] vNS2
(2)
Note that SI units should be used for the above correlation. More recently, Taitel and Dukler (1977) have developed a more complex model for frequency in horizontal, gasliquid slug flow. Their dimensionless frequency is dependent on five dimensionless groups, and a closed analytical form of correlation is not given. Therefore, because of its simplicity and accuracy, the Heywood-Richardson correltaion was used in estimating the frequency that was subsequently used in a dimensionless form in correlation of pipe top heat-transfer coefficients. Because the Dittus-Boelter correlation is more familiar and simple in form, the two-phase heat-transfer correlations are formulated to have a similar form. The dimensionless correlations developed for pipe bottom, pipe top, and overall average Nusselt numbers respectively are given in eqs 3-5. Nub = 0.023(ReL)o.s3PrLo.4( VsL/ VNS - 0.1)0.3 (3) Nut = 1.93Rem0.44Pr m0*4(Vs L/ vNS)0*21(~vSL/g)o*63(4) Nu,, = 0.023Re,0.s3Prm0.4( VsL/ vNS)o"6 (5) Here, ReL is the liquid Reynolds number based on the mixture velocity and PrLis the liquid Prandtl number: R ~ L D ~ N S P L / P L P ~ =L CPLPL/~L (6) Re, and Prm used in the correlation of pipe top and overall average Nusselt numbers are mixture Reynolds and Prandtl number respectively and are defined as Rem = DVNSPm/Pm fim = CPmPm/km (7) where Pm PLRS+ PO(^ - Rs) (8) Pm = PI,& + P G ( ~- Rs) (9) k, = kJis + k c ( 1 - Rs) (10)
In all the property correlations, Rs is the average liquid holdup in a liquid slug and is calculated by using the
6-
4-
5 E
d
m
3
z
2-
0.4
0.6
0.8 1.0 0.4
Prm
VSL
2.0 076
Figure 11. Correlation for overall average heat-transfer coefficients.
Heywood and Richardson (1979) correlation. It should be noted that the effect of gas-phase properties on the mixture properties is small and can be neglected in preliminary estimations of the Nusselt numbers. All three correlations are shown in Figures 9-11 respectively. It is evident that 93.5% of the data for pipe bottom and overall average heat-transfer Coefficients are correlated within a 110% range. For the pipe top heattransfer coefficients, 90% of the data fall within 115% of the correlated values. The pipe bottom Nusselt numbers showed a small inverse dependence on slug frequency. However, when used in the correlation as a dimensionless group, ita exponent was very small. Therefore, the dimensionless frequency group was not included in the correlation presented here (eq 3). As expected, the correlation for pipe top Nusselt number shows greater dependence on dimensionless frequency group. Since frequency is included in the correlation in a dimensionless group, ita effect on the heat transfer a t pipe top is combined with the effect of other parameters. Therefore, the pipe top Nusselt number is not directly dependent on frequency but is proportional
2178 Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991 Table IV. Comparison with Data Reported by Shoham (1982)
2.95 4.46 1.51 2.95 4.46 1.51 2.95 4.46 1.51 2.95 4.46
0.60 0.80
1.00
5481 4474 7913 8007 8147 7632 10845 9920 9942 9972 9996
4495 b 6047 6936 6322 7114 8424 8844 8078 9636 10493
nCalculated from the flow rates and temperatures reported in their paper. VsL/VNs is less than 0.1.
*
Table V. Comparison with Other Work D, VSL, VSG, h,&xpt), h, (predicted), %/m20~ m m/s m/s W/m20~ Fried’s (1954)Data and Predictions of Eq 5 0.019 4.66 4.80 25325 25906 6.06 4.12 26404 31733 0.019 0.019 6.06 6.48 24870 32246 Data from This Study and Lunde’s (1961) Correlation 0.028 0.30 1864 1681 0.30 0.028 0.62 3.07 3636 5354 0.028 1.02 2.01 6010 5290 0.057 0.20 1.15 1366 2007 0.057 0.30 0.39 1934 1484 0.067 0.60 2996 2654 1.22
to Moreover, the heat-transfer coefficients at pipe top show greater inverse dependence on pipe diameter. This is consistent with lower slug frequency and greater phase segregation in a larger pipe. Comparison with Other Work Only Mehta (1984) and Shoham et al. (1982) have reported measurements of pipe top and pipe bottom heattransfer coefficients for air-water flow. Mehta used a large (0.171 m) diameter pipe and reported only a few measurements. Since the measurements reported by Mehta (1984) are out of the range of data collected in this work, and LID ratio is very small, a comparison of our results with his measurements was not made. Comparison of Shoham et al. (1982) data of experimental heat-transfer coefficients at pipe bottom with predictions of our correlation is given in Table IV. It can be seen that the experimental values are somewhat underpredicted by the correlation. However, the agreement between the data and the predictions is fairly good. Table V shows a comparison between the experimental data for overall average heat-transfer coefficients reported by Fried (1954) and the correlation for haVs.developedin this study. In spite of the fact that the expenmental liquid and gas velocities are high, a close agreement with correlation is obtained. A similar comparison between some of the experimental data of this study with the predictions of the correlation developed by Lunde (1961) for overall average heat transfer coefficient is also summarized in Table V. An excellent agreement is obtained between the experimental and predicted values. Conclusions The steady-state heat-transfer rate in horizontal, plugslug flow varies around the pipe periphery. During this study, differences between pipe top and pipe bottom wall temperatures as high as 50 K were recorded. Therefore,
spatial average heat-transfer coefficient is not a true measure of the local heat-transfer phenomena. A knowledge of local phenomena is extremely important as variation in reaction rates caused by nonuniform temperatures would adversely affect the product quality. The heat-transfer coefficient at the pipe bottom was always higher than at the pipe top in a range of parameters studied. During two-phase flow, there was a reduction in heat transfer at pipe top as opposed to an enhancement at pipe bottom over the single-phase flow situation. Slug flow parameters such as frequency, mixture velocity, and the ratio of liquid or gas slug length to total length of slug unit affect the heat transfer and should be included when correlations are desired. Acknowledgment We gratefully acknowledge the financial support provided by the Chemical and Process Engineering Division of the National Science Foundation. The technical help received from the Chemical and Petroleum Engineering Department and the Mechanical Engineering Department at the University of Pittsburgh during equipment construction and operation is also appreciated. Appendix Data Reduction. Since variations in tube wall temperature around the circumference as well as an increase in wall temperature along the axis were noted, a variation in heat flux is expected. However, large radial variations of temperature did not occur because thin-walled test sections were used (pipe wall thicknesses were 0.002 m for the 0.057-m pipe and 0.003 m for the 0.028-in. pipe). Nevertheless, one method used to calculate the inside tube wall surface temperatures and the local heat flux that considered peripheral conduction around the tube wall (Schmidt and Sparrow (1978)) is outlined briefly. The measured circumferential outer wall temperatures Two were fitted to the expression T = T,
+ (To - T,) (l+,,,y.
(12)
where To and T, are the outer wall temperatures at Oo and 180° radial positions, respectively. From this, an average value of (a2T/a42)was obtained and used in the heat conduction equation
where Q is the heat generated per unit volume. Integrating and using aT/ar = 0 and T = Tw0(4)at r = r, (where r, is the outer radius) give
Iterating for Ti using the average of the inner and outer values of a2T/&h2did not change Ti(4)significantly. The local heat flux q(4) was determined from the first integration of eq 13 given as
where t is the wall thickness. The other two methods used to calculate the inside tube wall temperature from the measured outside tube wall
Ind. Eng. Chem. Res., Vol. 30,No. 9, 1991 2179
temperaturea and the average heat flux involve (1) solution of the simple heat conduction equation neglecting the thermal conductivity and electrical resistivity dependence on temperature for stainless steel and (2) series solution obtained where variation of electrical resistivity and thermal conductivity of stainless steel is considered (Bishop (1974); Kreith and Summerfield (1949)). Inside tube wall temperatures and heat fluxes calculated by using the above mentioned methods agreed within f5%. However, the local heat-transfer coefficients and Nusselt numbers were calculated as usual as
i = inside wall L = liquid m, mix = mixture NS = no slip (i.e., mixture) ?r = at 1 80' position p r = predicted S = slug (liquid) SG = superficial gas . SL = superficial liquid SP = single phase t = pipe top TP = two-phase TPB = two-phase, pipe bottom TPT = two-phase, pipe top = wall WO = outside wall x = at a n y axial position
w
Thus, steady-state local heat-transfer coefficients were calculated at the top of the tube, on the sides of the tube, and at the bottom of the tube. The circumferential averaged heat-transfer coefficients and Nusselt numbers were determined b y averaging the axial-average values at the bottom, top, and two sides of the pipe for each run. Error Estimation. The heat flux q is given by q = Q/(xDiL). Since the worst agreement for heat balances was *7% , by using a standard error analysis procedure, an estimated maximum error in heat flux values would be f7.9%. Now, the heat-transfer coefficient h is given b y h = q/(Ti- Tb). Thus,h is relatively insensitive to errors in measured temperatures provided that the difference Ti - Tbis large. Since the difference Ti - Tbat pipe top was always large, the maximum error in measured heabtransfer coefficients at pipe top is estimated to be k9.3%. A corresponding maximum error in measured pipe bottom heat transfer coefficients would be f11.5%. Nomenclature A = area C = constant
C p = specific heat
D = diameter g = acceleration due to gravity 4 = heat-transfer coefficient h = average heat-transfer coefficient k = thermal conductivity
L = length
m = constant in eq 12
Nu = Nusselt number
Nu
= average Nusselt number
Pr = Prandtl number
9 = heat generated per unit volume
Q = rate of heat transfer
q = heat flux Q = average heat flux R = holdup r = radius or radial position Re = Reynolds number
T = temperature
t = pipe
wall thickness V = velocity Creek Symbols = slug frequency 4 = angle defining circumferential location p = density Y
p
= viscosity
Subscripts avg = average b = bubble or bulk fluid or pipe bottom DB = Dittus-Boelter exp = experimental G = gas
Literature Cited Allen, R. W.; Eckert, E. R. G. Friction and Heat-Transfer Measurements to Turbulent Pipe Flow of Water (Pr= 7 and 8) at Uniform Wall Heat Flux. J. Heat Transfer 1964,86,301-310. Barnea, D.; Brauner, N. Holdup of the Liquid Slug in Two-Phase Intermittent Flow. Znt. J. Multiphase Flow 1985,11,43-49. Bishop, A. A. Crystallite Induction Time for Calcium Sulfate on a Heated Surface During Single-phase and Subcooled Nucleate Boiling Flows. Ph.D. Dissertation, Carnegie-Mellon University, 1974. Bishop, A. A,; Deshpande, S. D. Heat Transfer to Non-Newtonian Liquid-Gas Mixtures Flowing through Horizontal Tubes. Proceedings of the 8th International Heat Transfer Conference,San Francisco, August 1986;Vol. 3, pp 943-949. Deshpande, S. D. Study of Hydrodynamics and Heat Transfer in Non-Newtonian Liquid-Gas Two-Phase Flow in Horizontal Pipes. PbD. Dissertation, University of Pittsburgh, 1985. Deshpande, S. D.; Bishop, A. A. Heat Transfer to Non-Newtonian Liquids Flowing through Horizontal Tubes. Chem. Eng. Commun. 1987,52,339-354. Dukler, A. E.; Hubbard, M. G. A Model for Gas-Liquid Slug Flow in Horizontal and Near Horizontal Tubes. Znd. Eng. Chem. Fundam. 1975,14,337-347. Fedotkin, I. M.; Zarudnev, L. P. Correlation of Experimental Data on Local Heat Transfer in Heating of Air-Liquid Mixtures in Pipes. Heat Transfer-Sou. Res. 1970,2(1),175-181. Fried, L. Pressure Drop and Heat Transfer for Two-Phase, TwoComponent Flow. AIChE Symp. Ser. 1954,50(9),47-51. Gregory, G. A.; Scott, D. S. Correlation of Liquid Slug Velocity and Frequency in Horizontal Cocurrent Gas-Liquid Slug Flow. AIChE J . 1969,15,933-935. H e y w d , N. I.; Richardson, J. F. Slug Flow of Air-Water Mixtures in a Horizontal Pipe: Determination of Liquid Holdup by y-Ray Absorption. Chem. Eng. Sci. 1979,34,17-30. Hughmark, G. A. Holdup and Heat Transfer in Horizontal Slug Gas-Liquid Flow. Chem. Eng. Sci. 1965,20,1007-1010. Johnson, H. A.; Abou-Sabe, A. H. Heat Transfer and Pressure Drop for Turbulent Flow of Air-Water Mixtures in a Horizontal Pipe. Trans. ASME 1952,74,977-987. Kago, T.; Saruwatari, T.; Kashima, M.; Morooka, S.; Kato, Y. Heat Transfer in Horizontal Plug and Slug Flow for Gas-Liquid and Gas-Slurry Systems. J. Chem. Eng. Jpn. 1986,19(2), 125-131. Kreith, F.; Summerfield, M. Heat Transfer to Water at High Flux Densities With and Without Surface Boiling. Trans. ASME 1949, 71,805-814. Lunde, K. E.Heat Transfer and Pressure Drop in Two-Phase Flow. Chem. Eng. B o g . Symp. Ser. 1961,57,104-110. McAdams, W. H. Heating and Cooling Inside Tubes. In Heat Transmission, 3rd ed.; McGraw-Hill: New York, 1954;p 219. Mehta, D. C. 'Full-scale Cold-Flow Modeling of the SRC-I Slurry Fired Heater at Creare, Inc." DOE Report No. DOE/OR/ 03054-58(DE84013792),1984. Oliver, D. R. Wright, S. J. Pressure Drop and Heat Transfer in Gas-Liquid Slug Flow in Horizontal Tubes. Br. Chem. Eng. 1964, 9(9),590-596. Petukhov, B. S. Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties. Ado. Heat Transfer 1970,6, 504-564. Pletcher, R. H.; McManue, H.N. Heat Transfer and Preseure Drop in Horizontal Annular Two-Phase, Two-Component Flow. Int. J.
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Heat Mass Transfer 1968,11, 1087-1104. Schmidt, R. R.; Sparrow, E. M.Turbulent Flow of Water in a Tube with Circumferentially Nonuniform Heating, With or Without Buoyancy. J. Heat Transfer 1978,100,403. Shoham, 0.;Dukler, A. E.; Taitel, Y. Heat Transfer During Intermittent/Slug Flow in Horizontal Tubes. Znd. Eng. Chem. Fundam. 1982,21,312-318. Taitel, Y.; Dukler, A. E. A Model for Slug Frequency During Gas-
Liquid Flow in Horizontal and Near Horizontal Pipee. Znt. J. Multiphase Flow 1977,3,585-596. Tronconi, E. Prediction of Slug Frequency in Horizontal Two-Phase Slug Flow. AZChE J. 1990,36(5), 701-709.
Received for review October 12, 1990 Revised manuscript received January 14, 1991 Accepted February 27,1991
Prediction of High-pressure Gas Solubilities in Aqueous Mixtures of Electrolytes Kim Aasberg-Petersen, Erling Stenby,* and Aage Fredenslund Engineering Research Center IVC-SEP, Imtitut for Kemiteknik, The Technical University of Denmark, DK 2800 Lyngby, Denmark
This work presents a new model for prediction of the solubilities of gases in aqueous electrolyte solutions at high pressures. The model combines an equation of state (EOS)with a modified Debye-Huckel electrostatic contribution. The EOS used in this work is the ALS equation with a volume-dependent mixing rule for determination of the mixture a parameter. A single temperatureand composition-independent parameter is used to take into account gas-salt interactions. This parameter may be obtained by using readily available low pressure experimental data. Predicted solubilities of N2, CHI, and C02 in aqueous solutions of NaCl and CaC12agree well with experimental data at almost all pressures and salt concentrations up to 4 mol/kg. The new model is compared with and found to be superior to an earlier model of Harvey and Prausnitz. Finally, the solubility of a natural gas in a reservoir brine has been calculated. The predicted solubilities are in good agreement with experimental data. Introduction The purpose of this work was to develop a model for predicting the high-pressure solubility of natural gases in brine or formation water. The mixtures of interest contain various light hydrocarbons, Nz, COP,and water with dissolved salts, often at high temperatures and pressures. Many models have been developed that are able to predict phase equilibria of mixtures that contain both polar and nonpolar compounds. This is usually done either by introducing a volume-dependent a parameter mixing rule into a conventional cubic equation of state (EOS) or by coupling the EOS with an activity coefficient model (e.g., Huron and Vidal, 1979; Mathias and Copeman, 1983; Mollerup, 1985; Gani et al., 1989; Michelsen, 1989). However, none of these models can take into account the presence of ions. Numerous models have also been developed for predicting vapor-liquid equilibria in mixtures containing electrolytes (e.g., Sander et al., 1986; Macedo et al., 1990) but do not in general include noncondensable gases. In addition, these models are usually limited to low pressures. It is the purpose of this paper to describe a new model for predicting the high-pressure gas solubilities in aqueous electrolyte solutions. It was our objective to develop a simple method for which the necessary interaction parameters for high-pressure predictions can be obtained by using easily available low-pressure experimental data.
The Model The model described in this paper is based on the assumption that no ions are present in the gas phase. It is therefore necessary only to develop expressions for calculation of the fugacities of the nonionic components in the mixture. The following equation has proven to be a successful1 basis for predicting vapor-liquid equilibria for mixtures
with electrolytes at low pressures (Sander et al., 1986; Macedo et al., 1990): In yi = In y:CT + In yiEL (1) yiACTis evaluated by using a conventional activity coefficient model (e.g., UNIQUAC), and rimis an electrostatic contribution. It is not possible to use activity coefficient models for mixtures with noncondensable gases at high pressures. The following expression analogous to eq 1 is therefore suggested as basis for the new model: In di = In diEoS+ In 4iEL i = 1, ...,N (2)
N is the number of nonelectrolytic components (gases or solvents) in the mixture. Equation 2 states that the fugacity coefficient may be calculated as a product of a contribution from an EOS and a contribution from an electrostatic term. The second term on the right-hand side of eq 2 is equal to zero if the mixture is free of electrolytes. Since di = di0yi,eq 2 may be rewritten as In 4i = In 4io+ In yiEoS+ In yPL i = 1, ...,N (3) di0 is the pure component fugacity coefficient at the same temperature and pressure as the mixture. Since $iois independent of composition, dio is equal to 4j0ms.Equation 3 then becomes In di = In dims + In yiEL i = 1, ...,N (4) Any model suitable for correlation of gas-water equilibria may be used to calculate the first term on the right-hand side of eq 4. In this work the ALS EOS (Adachi et al., 1983) as modified by Jensen (1987) is employed. The ALS EOS is a(T) p = - RT (5) u - bl ( u - b&(U + b,)
0888-5885f 91f 2630-218O$02.50f 0 0 1991 American Chemical Society
