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Heat transfer in bubble layers at high pressures
Heat Transfer in Bubble Layers at High Pressures A. A. Avdeev B. F. Balunov V. I. Kiselev All-Union Nuclear Power R & D Institute (VNIIAM), Moscow, Russia
• An original experimental investigation of heat transfer with steam condensation on a surface of a horizontal cooled tube immersed in a bubbling layer was carried out. A copper test section 16 mm in diameter and 285 mm in length was placed in a bubbling column 295 mm in diameter. Experiments were made under a pressure of 0.72-3.8 MPa with volume steam content 0-0.18, steam superficial velocities 0-0.18 m / s , and liquid-wall temperature difference 38-106 K. The heat transfer process in a bubbling layer under high pressures is shown to be of considerably intensity; with moderate values of steam content heat transfer coefficients reach 10-12 k W / ( m 2. K). The use of the known correlations assumed for the case of air bubbling under atmospheric pressure results in systematically underestimating heat transfer by 30-80%. Data were obtained on heat transfer with film condensation of steam and natural convection of subcooled water at high temperature differences outside the range investigated earlier. Experimental data table is appended. Keywords: bubbling, high-pressure steam, heat transfer, horizontal tube, cooling, condensation
INTRODUCTION
EXPERIMENTAL APPARATUS
The problem on determining heat transfer from heat transfer surfaces immersed in a gas-bubbled liquid volume occurs in many engineering applications. Because of the importance of the problem a considerable number of experimental data on heat transfer in bubbling layers have already been gathered. H e a t transfer from bubbling column walls [1], single tubes [2], and vertical and horizontal tube bundles [3, 4] has been investigated. Experiments have been carried out with various liquids (ethanol, water, chlorinated carbon, machine oil, etc.) in a wide range of superficial gas velocities [5]. There are correlations generalizing the experimental data obtained [1, 4, 6-9]. The case of a gas (as a rule, air) bubbling at pressures close to atmospheric was studied in all the publications available to us. The only exception was a series of investigations made at the H e a t Physics Institute of the U S S R A c a d e m y of Sciences, Siberian branch, in which heat transfer coefficients from porous gas distribution plates placed at the bottom of the vessel, were measured. Experiments have shown that changes in gas pressure in a bubbling volume exert an appreciable influence on heat transfer. Thus the key purpose of the present work has been to experimentally investigate heat transfer in a bubbling layer at high pressure and compare the experimental data with the available correlations.
The top section of a large-scale experimental installation to test natural circulation was used as a bubbling vessel. It was a cylindrical column of 295 mm I.D. and 4.04 m height (Fig. 1). Steam generated by an electric heater placed in the bottom of the installation was bubbled through a water layer and then discharged into a condenser with the discharge valve positioned on the upper cover of the installation. The steam supply section was 12 m lower than the test section. Such a considerable distance kept the radial steam content profile stable (according to the available experimental data, even with a nonuniform steam load the cross-sectional steam content distribution stabilizes at a distance of two to three column diameters from the steam supply level). The test section was positioned in the middle section of the column. During its construction we did our best to achieve a uniform temperature distribution along its length and perimeter. For this reason the test section was made of copper tubing 16 mm in outer diameter and 285 mm in length. The tube wall thickness was 3 mm. During the experiments operating conditions were such that within a channel, surface boiling of the cooling water occurred. To measure the bulk temperature of the cooling water, a mixer was mounted at a test section outlet. This was a cylindrical bushing filled with stainless steel chips. To reduce longitudinal heat losses from the ends of the test section, stainless steel adapters were welded on. The
Address correspondence to Dr. Alexandr A. Avdeev, All-Union Nuclear Power R & D Institute (VNIIAM), Cosmonavt Volkov Str., Moscow, 125171 Russia.
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Heat Transfer in Bubble Layers 729 adapters were separated from the bubbling volume by annular plates and had a Teflon coating 10 mm thick for thermal insulation. To suppress natural circulation and reduce the heat supply from the two-phase mixture to the adapters, cylindrical spaces confined by the annular plates were tightly filled in with an asbestos filament. The key measuring equipment arrangement is shown in Fig. 1. The bulk temperature of the cooling water at the test section inlet, Ti,, and outlet, Tout, the bubbling layer temperatures T u T 2 , and the temperature distribution along the test section length and perimeter, T11 - T TM, were recorded. In addition, the pressure Pl in the bubbling layer, the cooling water pressure P2, superficial steam velocity w 0 in a column, cooling water flow rate G, and true average volume steam content q~ over a column section were measured at a point in the test section. To measure temperatures, chromel-alumel cable thermocouples (thermocouple wire diameter 0.1 mm, outer thermocouple cable diameter 1 mm) were used. Hot junctions of thermocouples measuring the test section wall temperature (T) - T~v) were embedded in longitudinal rectangular grooves 1 x 1 mm in cross section and 15 mm in length, caulked in, and sealed by fusion with high-temperature silver solder and subsequent machining. Hot junctions of thermocouples recording operating media temperatures (T1, T2, Tin , To, t) were placed directly in the flow. Temperatures were recorded by means of single- and multipoint electronic potentiometers. Cooling water inlet and outlet temperature measurements were overlapped by indications of mercury thermometers graduated to 0.5 K. The level position of the two-phase mixture and the true volume steam content average over a column section were determined on the basis of measurements of hydrostatic pressure differences Ap in separate sections 1 m in length arranged along the column length. Figure 1 shows the arrangement of only one pair of pressure taps for measuring Ap in the test section location. The pressure drop recorded by a difference pressure gauge is related to average two-phase mixture density in
the column by the formula A p = gh( pp
-
Ptp)
(1)
This formula allows us to determine Ptp and consequently the steam content average over the section: = ( Pl -- Ptp)/( Pl -- Pv)
(2)
To improve the accuracy of the measurements, the following procedure was used. Before each series of experiments the installation was brought to the pressure desired and heated up to saturation temperature. In the no-boiling condition, a reading was made of the hydrostatic pressure difference Ap0 corresponding to zero steam content. Upon completion of a series of measurements, the level was decreased, and the pressure difference Apa corresponding to the case ~p = 1 was read. With the known Ap0 and Apl, the value of q~ was determined from the ratio ~p = ( A p - A p o ) / ( A p l
- Ap0 )
(3)
The cooling water flow rate G was determined with a volume method as
G = PoutV/t
(4)
where t is the time required to fill the vessel of volume V. The value of the superficial steam velocity w 0 was independently measured by two methods: by electrical heating power, taking into consideration heat losses into the environment that had been measured during a special series of experiments, and also by the rate of change in mass level of liquid in the installation, which had been determined by indications of pressure difference gauges Ap. In the first case, w0 = 4 ( N
-
Q)/(TrdZp~r),
(5)
!
,o I
I
•
-
~f f -
o
....
¢~95
/¢t
- -
-
-
o
Figure 1. Experimental installation. 1, copper test section; 2, mixer; 3, stainless steel adapters; 4, annular plates; 5, Teflon coating.
730 A.A. Avdeev et al. and in the second, wo
1
d(Ap)
Pvg
dt
The formula to calculate the relative measurement error of the flow rate follows from Eq. (4): (6)
Heat flux density on a test section surface and the heat transfer coefficient were calculated as q = CpG(Tou t - T i n ) / ( T r d t L t )
(7)
and a = q / ( T ~ - Tw)
(8)
respectively. MEASUREMENT E R R O R Gauge pressure in the bubbling column was measured by a reference pressure gauge within an accuracy of 0.35% and measurement limits 0-4 MPa. Correspondingly, the absolute error of pressure measurement 6 p was -+0.014 MPa. With the highest pressure (p = 3.9 MPa), this error resulted in a determination error for the saturation temperature of 6Ts = 0.2 K, while with the lowest pressure (p = 0.65 MPa), 6Ts = 0.9 K. Owing to the high thermal conductivity of the test section wall material, the error of surface temperature measurement relative to the temperature change over the wall can be neglected. Then the total measurement error with respect to temperature would be governed by the accuracy of thermocouple calibration and an error of the secondary facilities. Thermocouples were calibrated with an accuracy 6 T a = 0.3 K. An electronic potentiometer used to record temperatures had an accuracy of 0.25% and an upper measurement limit of 10 mW. Correspondingly, the recording error with respect to thermal emf was equal to -+0.025 mW, which results in a temperature measurement error 6TE = 0.63 K. The total error of temperature measurement is 6 T = 6 T E + 6 T G = 0.93 K
(9)
All temperature measurement channels were checked daily under isothermal conditions. During the check the cooling water supply through the test section was stopped and the liquid level in the installation was lowered to below the level of the test section. Temperatures measured under these conditions were compared with the saturation temperature determined from the pressure in the bubbling layer. Disagreements did not exceed + 1 K, which is close to Eq. (9). In making further estimations, 6 T was assumed to be equal to 1 K. To measure hydrostatic pressure differences, pressure difference gauges with an output electric current signal of 0-5 mA and an accuracy of 1.0% were used. Correspondingly, the reading error of pressure difference in terms of the output signal 6(Ap) was 0.05 mA. From Eq. (3), the absolute measurement error of the true volume steam content is 2a(Ap) 6~o = (1 + q~)Ap 1 _ Apo
(10)
With a maximum ~o of 0.2 and the magnitude of difference Apo - A p a = 4 mA, we obtain 8~o = 0.03.
6G
6V
6t + -- + V t
G
~Pout Pout
= 0.02
(11)
The error in the determination of superficial steam velocity from electric heater power in accordance with Eq. (5) was 6w 0
6N
w0
N - Q
+ 2
6d c ~
6po 6r + -- + -p~ r
(12)
The relative measurement error of electric heater power N was _+2%. Under high pressures ( p = 3.9 MPa) the heating power on average was 150 kW, and the value of heat losses into the environment, Q, was equal to 45 kW. The error of experimental determination of Q has been estimated to be +5%. Then, from Eq. (12) it follows that 8 W o / W o = 0.08. Under the lowest pressures (p = 0.75 MPa), N = 55 kW and Q = 35 kW, and results calculated with Eq. (12) show the considerably high value 6 W o / W o = 0.17. The error in determination of w0 from the change in the mass level of liquid in the installation is smaller. Under steady-state conditions, d ( A p ) / d t is constant. The value of this derivative was determined as the ratio of the pressure difference Ap at the beginning and end of the experiment to measurement time At. Considering this, from Eq. (6) under low pressures it follows that 6w o
6po
w0
po
+ 2
6(Ap)
6(At) + - At
Ap
0.13
(13)
At high pressures the determination error of w0 is somewhat lower. Under high pressures the superficial steam velocity was determined with Eq. (5), and under low pressures, with Eq. (6). A comparison of experimental values of w0 determined by these two procedures has given a disagreement on the order of + 5 % under high pressures and _+15% under low pressures. From Eq. (7) it follows that the maximum determination error of heat flux density is ~£p --+ ep
6q q
8G
6T
+
~
2 Tout _ Tin
(14) 6d t
+-~t
~L t
+ L-T =0"09
Similarly, from Eq. (8) it follows that the total determination error of the heat transfer coefficient does not exceed 6a a
6q
6T + 2 - q T~-T w
0.13
(15)
EXPERIMENTAL P R O C E D U R E The experimental installation was filled with chemically desalinated deaerated condensate, after which electric
Heat Transfer in Bubble Layers 731 heating was started. Warming up of the installation was accompanied by the release of steam through the discharge valve. After the temperature of the two-phase volume was stabilized, cooling water was supplied through the test section. The water flow rate was set in such a way that the outlet water temperature was within the range 55-75°C under an inlet pressure of 0.11-0.12 MPa. Thus, the value of the cooling water outlet bulk subcooling always exceeded 30 K. This maintained complete condensation in a mixer for the steam that was generated on the inner surface of the test section walls due to the presence of surface boiling. The absence of steam passing through the mixer was confirmed by coincidence of outlet thermocouple readings with the results of cooling water temperature measurements in a measuring vessel. By changing the heating power and steam release through the discharge valve, the installation was brought to the desired conditions. After the pressure and temperature were stabilized, readings of all of the measurement equipment were taken. Then the installation was changed over to a new set of conditions. In this manner a series of experiments were carried out under constant pressure and different steam loads. In the course of the experiments, because of water evaporation, there was a gradual decrease in the liquid level in the bubbling vessel. When the two-phase level was lowered to the same level as the upper pressure difference tap (325 mm from the test section arrangement point), fresh condensate make-up was made. This moment was determined by the beginning of the monotonic change in time of pressure difference gauge readings. RESULTS OF C O N T R O L E X P E R I M E N T S To verify the test procedure, two series of control experiments were carried out. These consisted of heat transfer measurements with one-phase natural convection of subcooled liquid and film condensation of saturated steam. The experiments on natural convection were conducted under pressures of 1.9-3.0 MPa, liquid subcooling 3.5-40.0
N~
I
1 - - [
K, liquid-wall temperature differences AT = 84-110 K, and heat flux densities q - 0.16-0.30 M W / m 2. The results of this series of experiments are presented in Fig. 2 plotted as Nu = f ( G r Pr). In Fig. 2 a line calculated with the recommendation of Ref. 11. Nu = 0.5(Gr Pr) °'25
is also presented. While processing data all of the properties were taken over the mean temperature of a boundary layer, which was equal to half the sum of the liquid and wall temperatures [11]. Despite the fact that data obtained are somewhat outside the applicability range of the semiempirical Eq. (16) (our data spread over the range 0.84 × 108 < Gr Pr < 1.7 × 108, while it is recommended that the relationship discussed be applied at GrPr < 108), the maximum disagreement does not exceed 14%, which is close to the experimental error. In all experiments of this series there was no systematic change in wall temperature over the test section length and perimeter. Experiments on film condensation were carried out with p = 2.0-3.7 MPa, AT = 78-105 K, and q = 0.68-0.89 M W / m 2. Superficial steam velocity during these experiments was within the range 0-0.09 m / s and did not influenced heat transfer. Figure 3 shows a comparison of the results obtained in the present work (the top cluster of points) with other available experimental data on the condensation of vapors of various substances (the bottom group of points) as well as with results of calculations using Labuntsov's formula [12]. It is seen that there is good agreement between these sets of data. The relationship [12] Ref
= 0.81Z3/4•
8O
J
o
] 6"
I 8
t /0 ~
(17)
is one of the most precise. It takes account of the variability of condensate properties, subcooling, and effects releated to liquid film wave motion.
I
60
(16)
I 2
Figure 2. Comparison of heat transfer measurements with water natural convection values calculated with Eq. (16).
732 A.A. Avdeev et al. r
I
/00 8 0 g-
2
tO ?3 6
J¢
4
I
2
I
4'
I
6
~ ~ /
I
a
I
//
I
6
I
I to
1 a2
I
4
I
6
I
I
/00
1
2
I
4/
I
G
I] ,~
Figure 3. Comparison of heat transfer measurements with film condensation values calculated with Eq. (17). ((3, data of previous works; A, data of the present work) During this series of experiments, a change in test section wall temperature over the perimeter caused by an increase in condensate film thickness was observed. However, the maximum change in the wall-liquid temperature difference over the perimeter did not exceed + 6% and did not exert any appreciable influence on heat transfer. It should be noted that the results of control experiments on natural convection and film condensation somewhat widen the range of parameters studied.
St = 0.1 (Re Fr Pr 2 ) - 025
DISCUSSION A series of experiments to measure heat transfer to a cooled cylindrical surface immersed in a saturated water layer through which steam is being bubbled were carried out in the following range of operating conditions: p = 0.72-3.8 MPa; w0 = 0-0.085 m/s; q~ = 0-0.18; q = 0.36-1.1 M W / m 2. The main results of these measurements are given in Table 1. Less than 10% of the total steam supplied to the bubbling vessel condensed on the test section surface. Therefore, in processing the data, we ignored the change in superficial steam velocity along the test section height. During experiments on bubbling heat transfer there was no systematic change in wall temperature over the test section length and perimeter. The arithmetic mean of time-averaged readings of all thermocouples recording wall temperature was used as a reference value. As steam content (superficial steam velocity) increased, the heat transfer coefficients monotonously increased (Fig. 4). On a logarithmic scale, the relationship a = f(w o) is well approximated by straight lines corresponding to power low
a = const ~n
well illustrated by Fig. 4 in which experimental points obtained at the lowest pressures (p = 0.72-0.76 MPa) and those obtained at the highest pressures (p = 3.53-3.83 MPa) are plotted. A comparison of measurement results with the known correlations that generalize experimental data obtained under atmospheric pressure shows systematic disagreement. As an example, Fig. 5 shows a comparison of our measurement results with the Deckwer correlation [6], (19)
as well as with other correlations [1, 8, 9]. It is seen that the known correlations result in a systematic underestimation of heat transfer by 30-80%. Several different explanations for these disagreements are possible. As we see it, a change in phase circulation hydrodynamics in the bubbling colum with growing pressure [13, 14], which is not taken into account by formulas like Eq. (19), is the most likely reason. PRACTICAL S I G N I F I C A N C E / U S E F U L N E S S High values of heat transfer coefficients that reach magnitudes on the order of 10-12 k W / ( m 2. K) with moderate steam content values attract our attention. These values somewhat exceed the values typical for film condensation. To maintain such considerable values of heat transfer coefficients with water, cross flow velocities on the order of 10 m / s are required. It seems to be reasonable to revise the commonly used approach to the design of built-in heat exchangers for natural circulation loops, locating the greater part of their surface below the coolant level in a bubbling layer.
(18)
where the exponent n falls within the limits of 0.25-0.29. There is a certain divergence between experimental points of different pressures in coordinates a =f(q~). This is
CONCLUSIONS An experiment was carried out to heat transfer with high-pressure steam condensation on a cooled surface
Heat Transfer in Bubble Layers i
8 8 7 6 # ,/,/
-
I
I
-~
I
733
I
/
m
¢
..
I
1
I
I
I
I
e
~
a
8
/o
ao
u2, 9;
Figure 4. Dependence of coefficient of heat transfer with bubbling on steam content.
Table 1. Measurement Results 1
2
Pressure (MPa)
Volume Steam Content (%)
0.755 0.755 0.755 0.775 0.775 0.746 0.730 1.18 1.18 1.26 1.29 1.32 2.19 2.24 2.28 2.28 2.28 2.33 2.33 2.85 2.97 2.80 2.94 3.24 2.70 3.06 2.89 2.70 2.75 2.75 3.53 3,73 3,73 3.83 3.83 3.78 3.53 3.53 3.58
2.0 3.0 4.0 5.7 7.3 8.5 10.5 7.5 8.2 9.8 11.0 12.8 5.0 6.8 9.3 10.5 12.7 13.4 13.4 3.0 3.5 3.8 7.0 7.5 8.7 9.0 13.3 14.0 14.8 15.3 7.2 10.0 13.0 14.5 16.5 18.0 15.0 17.5 18.0
3
Superficial Steam Velocity (m / s) -----0.060 0.070 ---0.060 0.070 --0.030 0.040 0.060 0.070 0.070 -0.023 0.023 0.021 0.023 0.040 -0.080 0.082 0.084 0.085 0.023 0.041 0.052 0.053 0.063 0.080 0.070 0.080 0.080
4
5
6
7
8
Inlet Water Temp. (°C)
Outlet Water Temp. (°C)
Wall Temp. (°C)
Water Flow Rate (g / s)
Heat Transfer Coeff. [kW / (me . K)]
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5,0 5,0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 24.0 24.0 24.0 24.0 20.5 5.0 20.5 20.5 20.5 20.5 24.0 24.0 24.0 24.0 24.0 24.0 5.0 5.0 5.0
36.0 52.0 58.0 62.0 63.0 64.5 66.5 61.0 63.0 65.0 66.0 68.0 59.0 62.0 64.0 67.0 55.0 57.0 58.0 64.0 71.7 69.3 78.3 61.7 68.5 60.0 76.5 75.5 76.5 78.0 61.7 65.5 67.9 70.3 70.8 71.1 48.5 54.0 51.5
114.0 119.0 122.5 124.0 125.5 127.0 128.0 128.0 130,0 131.0 133.0 133.0 134.0 143.0 143,0 143.0 144.0 145.0 146,0 130.0 133.5 132,0 137.5 139.0 134.0 133.0 141,0 140,0 142.0 142.0 139.0 140.0 145.0 148.0 148.0 149,0 144,0 146.0 146.5
42.5 26.7 26.5 26.7 26.2 25.8 25.7 35.6 36.0 38.3 38.6 37.3 46.3 46.3 45.0 44.7 59,7 58,3 58.7 37.6 59.0 59.0 58.5 81.0 61.0 55.0 61.0 61.0 61.0 61.0 81.0 81.0 81.0 81.0 81.0 81.0 82.7 81.3 81.7
7.12 7.46 9.00 9.85 10.2 11.1 12.2 9.85 10.7 11.4 11.8 11.6 8.80 10.3 10.2 10.6 11.6 11.8 12.2 6.47 8.23 7.96 9.73 8.99 9.09 8.66 11.0 11.1 11.5 11.8 8.56 9.24 10.3 11.0 11.0 11.1 10.6 12.0 11.4
734 A.A. Avdeev et al.
8~
- -
l
I
•
r
.........
] .....
q
-F--T--
I
~teiff~ l~e~spach P.N.{ g-7 0
/ - - / 4 a ~ WF'.{ ¢7 --
Q~
~o//
o,o~
o
o,o6 00
_
o,o~ A
0
0 0
°°°
-_
8o o
I
1
I
I
I
I
I
I
I
6
8
¢o
~0
ko
6o
~
¢oo
~oo
I~PeP~
Figure 5. Comparison of heat transfer measurements at bubbling (points) with calculations made using known formulas (lines).
~
immersed in a bubbling layer for the first time. The results of measurements are not described by the known semiempirical correlations constructed on the basis of data on heat transfer with gas bubbling at atmospheric pressure. To ascertain the causes of these disagreements further experimental studies are required in the pressure range 0.1-1.0 MPa.
Tout bulk cooling water temperature at test section outlet, K AT liquid-wall temperature difference, K t time, s V measuring vessel volume, m 3 wo superficial steam velocity in bubbling column, m / s Z parameter in Eq. (17) ( = Ga 1/3 h s AT/r~s), dimensionless
NOMENCLATURE
Greek Symbols a heat transfer coefficient, W / ( m 2. K) /3 temperature coefficient of liquid volume expansion [= (op#ar)JpA, K -1 • condensate properties variability correction, Eq. (17) 3 3 {[(A~/As)( IzJl~w)] 1/8 }, dimensionless A thermal conductivity of liquid, W / ( m . K) /z dynamic viscosity of liquid, N . s / m E v kinematic viscosity of liquid, m2/s p density, k g / m 3 Pout cooling water density at test section outlet, k g / m 3 lop water density at pressure taps, k g / m 3 Ptp two-phase mixture density [= pt(1 - ~) + Pv~], kg/m 3 true volumetric steam content averaged over column cross section, dimensionless
a thermal diffusivity of liquid (= A/ptCp), m / s Cp heat capacity of liquid at constant pressure, J / ( k g .
K) d c bubbling column diameter, m d b bubble diameter, m d t test section diameter, m Fr Froude number (= w~/gdb), dimensionless G cooling water flow rate, k g / s Ga Galilei number (= gH3/v~), dimensionless Gr Grashof number (= g/3ATd~/v2), dimensionless g gravitational acceleration, m / s 2 H linear scale, Eq. (17), m h vertical distance between pressure difference taps, m L t test section length, m N electric heater power, W Nu Nusselt number ( = ad/h), dimensionless Pr Prandtl number ( = v/a), dimensionless p pressure, Pa Ap hydrostatic pressure difference, Pa Q heat losses into the environment, W q heat flux density, W / m 2 Re Reynolds number ( = wodb/v), dimensionless Ref modified Reynolds film number {= qH/rlxs[1 +
(3/8)(Cp AT~r)]}, r St T Tin
specific heat of vaporization, J / k g Stanton number [(= q/( plWoCp AT)], dimensionless temperature, K bulk cooling water temperature at test section inlet, K
Subscripts
l s v w
liquid in saturation state vapor wall
REFERENCES 1. Hart, W. F., Heat Transfer in Bubble-Agitated System. A General Correlation, Ind. Eng. Chem. ProcessDes. Dev., 15, 109-114, 1976. 2. Burkel, W., Der Warmenbergang and Heiz- und Kuhlfachen in begasten Flussigkeiten, Chem. Ing. Techn.,5, 265-268, 1972. 3. Saxena, S. C., and Valdivel, R., Heat Transfer from a Tube Bundle in a Bubble Coumn, Int. Commun. Heat Mass Transfer, 15, 657-667, 1988.
H e a t Transfer in Bubble Layers 4. Tarat, E. Ya., Hoze, A. N., and Sharov, Yu. I., Investigation of Heat Transfer from a Tube Bundle in a Foam Layer, in Heat and Mass Transfer, Vol. 4, pp. 336-343, Academy of Science of BSSR, Minsk, 1968. 5. Sokolov, V. N., and Salamahin, A. D., Heat Transfer from Gas-Liquid System to a Heat Transfer Element Wall Under Bubbling Conditions, J. Appl. Chem., 35, 1022-1026, 1962. 6. Deckwer, W. D., On the Mechanism of Heat Transfer in Bubble Column Reactors, Chem. Eng. Sci., 35, 1341-1346, 1980. 7. Joshi, J. B., Sharma, M. M., and Shah, Y. T., Heat Transfer in Multiphase Contactors, Chem. Eng. Commun. 6, 257-271, 1980. 8. Kast, W., Analyse des Warmeubergangs in Blasensaulen, Int. J. Heat Mass Transfer, 5, 329-336, 1962. 9. Steiff, A., and Weinspach, P. M., Heat Transfer in Bubble Systems, German Chem. Eng., 1, 150, 1978. 10. Kutateladze, S. S., and Nakoryakov, V. E., Heat and Mass Transs-
735
fer and Waves in Gas-Liquid Systems, Nauka, Novosibirsk, 1984 (in Russian). 11. Grigoriev, V. A., and Zorin, V. M., Eds., Heat and Mass Transfer. Thermal Engineering Experiment: Handbook, Energoatomizdat, Moscow, 1982 (in Russian). 12. Labuntsov, D. A., Heat Transfer with Film Condensation of Pure Vapors on Vertical Surfaces and on Horizontal Tubes, Thermal Eng., 7, 72-79, 1957. 13. Avdeev, A. A., Hydrodynamics of Bubbling, Thermal Eng., 11, 42-46, 1983. 14. Avdeev, A. A., and Sirenko, E. I., The True Volume Steam Content with Bubbling, Thermal Eng., 11, 43-46, 1984.
Received April 1, 1991; revised February 12, 1992
• An original experimental investigation of heat transfer with steam condensation on a surface of a horizontal cooled tube immersed in a bubbling layer was carried out. A copper test section 16 mm in diameter and 285 mm in length was placed in a bubbling column 295 mm in diameter. Experiments were made under a pressure of 0.72-3.8 MPa with volume steam content 0-0.18, steam superficial velocities 0-0.18 m / s , and liquid-wall temperature difference 38-106 K. The heat transfer process in a bubbling layer under high pressures is shown to be of considerably intensity; with moderate values of steam content heat transfer coefficients reach 10-12 k W / ( m 2. K). The use of the known correlations assumed for the case of air bubbling under atmospheric pressure results in systematically underestimating heat transfer by 30-80%. Data were obtained on heat transfer with film condensation of steam and natural convection of subcooled water at high temperature differences outside the range investigated earlier. Experimental data table is appended. Keywords: bubbling, high-pressure steam, heat transfer, horizontal tube, cooling, condensation
INTRODUCTION
EXPERIMENTAL APPARATUS
The problem on determining heat transfer from heat transfer surfaces immersed in a gas-bubbled liquid volume occurs in many engineering applications. Because of the importance of the problem a considerable number of experimental data on heat transfer in bubbling layers have already been gathered. H e a t transfer from bubbling column walls [1], single tubes [2], and vertical and horizontal tube bundles [3, 4] has been investigated. Experiments have been carried out with various liquids (ethanol, water, chlorinated carbon, machine oil, etc.) in a wide range of superficial gas velocities [5]. There are correlations generalizing the experimental data obtained [1, 4, 6-9]. The case of a gas (as a rule, air) bubbling at pressures close to atmospheric was studied in all the publications available to us. The only exception was a series of investigations made at the H e a t Physics Institute of the U S S R A c a d e m y of Sciences, Siberian branch, in which heat transfer coefficients from porous gas distribution plates placed at the bottom of the vessel, were measured. Experiments have shown that changes in gas pressure in a bubbling volume exert an appreciable influence on heat transfer. Thus the key purpose of the present work has been to experimentally investigate heat transfer in a bubbling layer at high pressure and compare the experimental data with the available correlations.
The top section of a large-scale experimental installation to test natural circulation was used as a bubbling vessel. It was a cylindrical column of 295 mm I.D. and 4.04 m height (Fig. 1). Steam generated by an electric heater placed in the bottom of the installation was bubbled through a water layer and then discharged into a condenser with the discharge valve positioned on the upper cover of the installation. The steam supply section was 12 m lower than the test section. Such a considerable distance kept the radial steam content profile stable (according to the available experimental data, even with a nonuniform steam load the cross-sectional steam content distribution stabilizes at a distance of two to three column diameters from the steam supply level). The test section was positioned in the middle section of the column. During its construction we did our best to achieve a uniform temperature distribution along its length and perimeter. For this reason the test section was made of copper tubing 16 mm in outer diameter and 285 mm in length. The tube wall thickness was 3 mm. During the experiments operating conditions were such that within a channel, surface boiling of the cooling water occurred. To measure the bulk temperature of the cooling water, a mixer was mounted at a test section outlet. This was a cylindrical bushing filled with stainless steel chips. To reduce longitudinal heat losses from the ends of the test section, stainless steel adapters were welded on. The
Address correspondence to Dr. Alexandr A. Avdeev, All-Union Nuclear Power R & D Institute (VNIIAM), Cosmonavt Volkov Str., Moscow, 125171 Russia.
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Heat Transfer in Bubble Layers 729 adapters were separated from the bubbling volume by annular plates and had a Teflon coating 10 mm thick for thermal insulation. To suppress natural circulation and reduce the heat supply from the two-phase mixture to the adapters, cylindrical spaces confined by the annular plates were tightly filled in with an asbestos filament. The key measuring equipment arrangement is shown in Fig. 1. The bulk temperature of the cooling water at the test section inlet, Ti,, and outlet, Tout, the bubbling layer temperatures T u T 2 , and the temperature distribution along the test section length and perimeter, T11 - T TM, were recorded. In addition, the pressure Pl in the bubbling layer, the cooling water pressure P2, superficial steam velocity w 0 in a column, cooling water flow rate G, and true average volume steam content q~ over a column section were measured at a point in the test section. To measure temperatures, chromel-alumel cable thermocouples (thermocouple wire diameter 0.1 mm, outer thermocouple cable diameter 1 mm) were used. Hot junctions of thermocouples measuring the test section wall temperature (T) - T~v) were embedded in longitudinal rectangular grooves 1 x 1 mm in cross section and 15 mm in length, caulked in, and sealed by fusion with high-temperature silver solder and subsequent machining. Hot junctions of thermocouples recording operating media temperatures (T1, T2, Tin , To, t) were placed directly in the flow. Temperatures were recorded by means of single- and multipoint electronic potentiometers. Cooling water inlet and outlet temperature measurements were overlapped by indications of mercury thermometers graduated to 0.5 K. The level position of the two-phase mixture and the true volume steam content average over a column section were determined on the basis of measurements of hydrostatic pressure differences Ap in separate sections 1 m in length arranged along the column length. Figure 1 shows the arrangement of only one pair of pressure taps for measuring Ap in the test section location. The pressure drop recorded by a difference pressure gauge is related to average two-phase mixture density in
the column by the formula A p = gh( pp
-
Ptp)
(1)
This formula allows us to determine Ptp and consequently the steam content average over the section: = ( Pl -- Ptp)/( Pl -- Pv)
(2)
To improve the accuracy of the measurements, the following procedure was used. Before each series of experiments the installation was brought to the pressure desired and heated up to saturation temperature. In the no-boiling condition, a reading was made of the hydrostatic pressure difference Ap0 corresponding to zero steam content. Upon completion of a series of measurements, the level was decreased, and the pressure difference Apa corresponding to the case ~p = 1 was read. With the known Ap0 and Apl, the value of q~ was determined from the ratio ~p = ( A p - A p o ) / ( A p l
- Ap0 )
(3)
The cooling water flow rate G was determined with a volume method as
G = PoutV/t
(4)
where t is the time required to fill the vessel of volume V. The value of the superficial steam velocity w 0 was independently measured by two methods: by electrical heating power, taking into consideration heat losses into the environment that had been measured during a special series of experiments, and also by the rate of change in mass level of liquid in the installation, which had been determined by indications of pressure difference gauges Ap. In the first case, w0 = 4 ( N
-
Q)/(TrdZp~r),
(5)
!
,o I
I
•
-
~f f -
o
....
¢~95
/¢t
- -
-
-
o
Figure 1. Experimental installation. 1, copper test section; 2, mixer; 3, stainless steel adapters; 4, annular plates; 5, Teflon coating.
730 A.A. Avdeev et al. and in the second, wo
1
d(Ap)
Pvg
dt
The formula to calculate the relative measurement error of the flow rate follows from Eq. (4): (6)
Heat flux density on a test section surface and the heat transfer coefficient were calculated as q = CpG(Tou t - T i n ) / ( T r d t L t )
(7)
and a = q / ( T ~ - Tw)
(8)
respectively. MEASUREMENT E R R O R Gauge pressure in the bubbling column was measured by a reference pressure gauge within an accuracy of 0.35% and measurement limits 0-4 MPa. Correspondingly, the absolute error of pressure measurement 6 p was -+0.014 MPa. With the highest pressure (p = 3.9 MPa), this error resulted in a determination error for the saturation temperature of 6Ts = 0.2 K, while with the lowest pressure (p = 0.65 MPa), 6Ts = 0.9 K. Owing to the high thermal conductivity of the test section wall material, the error of surface temperature measurement relative to the temperature change over the wall can be neglected. Then the total measurement error with respect to temperature would be governed by the accuracy of thermocouple calibration and an error of the secondary facilities. Thermocouples were calibrated with an accuracy 6 T a = 0.3 K. An electronic potentiometer used to record temperatures had an accuracy of 0.25% and an upper measurement limit of 10 mW. Correspondingly, the recording error with respect to thermal emf was equal to -+0.025 mW, which results in a temperature measurement error 6TE = 0.63 K. The total error of temperature measurement is 6 T = 6 T E + 6 T G = 0.93 K
(9)
All temperature measurement channels were checked daily under isothermal conditions. During the check the cooling water supply through the test section was stopped and the liquid level in the installation was lowered to below the level of the test section. Temperatures measured under these conditions were compared with the saturation temperature determined from the pressure in the bubbling layer. Disagreements did not exceed + 1 K, which is close to Eq. (9). In making further estimations, 6 T was assumed to be equal to 1 K. To measure hydrostatic pressure differences, pressure difference gauges with an output electric current signal of 0-5 mA and an accuracy of 1.0% were used. Correspondingly, the reading error of pressure difference in terms of the output signal 6(Ap) was 0.05 mA. From Eq. (3), the absolute measurement error of the true volume steam content is 2a(Ap) 6~o = (1 + q~)Ap 1 _ Apo
(10)
With a maximum ~o of 0.2 and the magnitude of difference Apo - A p a = 4 mA, we obtain 8~o = 0.03.
6G
6V
6t + -- + V t
G
~Pout Pout
= 0.02
(11)
The error in the determination of superficial steam velocity from electric heater power in accordance with Eq. (5) was 6w 0
6N
w0
N - Q
+ 2
6d c ~
6po 6r + -- + -p~ r
(12)
The relative measurement error of electric heater power N was _+2%. Under high pressures ( p = 3.9 MPa) the heating power on average was 150 kW, and the value of heat losses into the environment, Q, was equal to 45 kW. The error of experimental determination of Q has been estimated to be +5%. Then, from Eq. (12) it follows that 8 W o / W o = 0.08. Under the lowest pressures (p = 0.75 MPa), N = 55 kW and Q = 35 kW, and results calculated with Eq. (12) show the considerably high value 6 W o / W o = 0.17. The error in determination of w0 from the change in the mass level of liquid in the installation is smaller. Under steady-state conditions, d ( A p ) / d t is constant. The value of this derivative was determined as the ratio of the pressure difference Ap at the beginning and end of the experiment to measurement time At. Considering this, from Eq. (6) under low pressures it follows that 6w o
6po
w0
po
+ 2
6(Ap)
6(At) + - At
Ap
0.13
(13)
At high pressures the determination error of w0 is somewhat lower. Under high pressures the superficial steam velocity was determined with Eq. (5), and under low pressures, with Eq. (6). A comparison of experimental values of w0 determined by these two procedures has given a disagreement on the order of + 5 % under high pressures and _+15% under low pressures. From Eq. (7) it follows that the maximum determination error of heat flux density is ~£p --+ ep
6q q
8G
6T
+
~
2 Tout _ Tin
(14) 6d t
+-~t
~L t
+ L-T =0"09
Similarly, from Eq. (8) it follows that the total determination error of the heat transfer coefficient does not exceed 6a a
6q
6T + 2 - q T~-T w
0.13
(15)
EXPERIMENTAL P R O C E D U R E The experimental installation was filled with chemically desalinated deaerated condensate, after which electric
Heat Transfer in Bubble Layers 731 heating was started. Warming up of the installation was accompanied by the release of steam through the discharge valve. After the temperature of the two-phase volume was stabilized, cooling water was supplied through the test section. The water flow rate was set in such a way that the outlet water temperature was within the range 55-75°C under an inlet pressure of 0.11-0.12 MPa. Thus, the value of the cooling water outlet bulk subcooling always exceeded 30 K. This maintained complete condensation in a mixer for the steam that was generated on the inner surface of the test section walls due to the presence of surface boiling. The absence of steam passing through the mixer was confirmed by coincidence of outlet thermocouple readings with the results of cooling water temperature measurements in a measuring vessel. By changing the heating power and steam release through the discharge valve, the installation was brought to the desired conditions. After the pressure and temperature were stabilized, readings of all of the measurement equipment were taken. Then the installation was changed over to a new set of conditions. In this manner a series of experiments were carried out under constant pressure and different steam loads. In the course of the experiments, because of water evaporation, there was a gradual decrease in the liquid level in the bubbling vessel. When the two-phase level was lowered to the same level as the upper pressure difference tap (325 mm from the test section arrangement point), fresh condensate make-up was made. This moment was determined by the beginning of the monotonic change in time of pressure difference gauge readings. RESULTS OF C O N T R O L E X P E R I M E N T S To verify the test procedure, two series of control experiments were carried out. These consisted of heat transfer measurements with one-phase natural convection of subcooled liquid and film condensation of saturated steam. The experiments on natural convection were conducted under pressures of 1.9-3.0 MPa, liquid subcooling 3.5-40.0
N~
I
1 - - [
K, liquid-wall temperature differences AT = 84-110 K, and heat flux densities q - 0.16-0.30 M W / m 2. The results of this series of experiments are presented in Fig. 2 plotted as Nu = f ( G r Pr). In Fig. 2 a line calculated with the recommendation of Ref. 11. Nu = 0.5(Gr Pr) °'25
is also presented. While processing data all of the properties were taken over the mean temperature of a boundary layer, which was equal to half the sum of the liquid and wall temperatures [11]. Despite the fact that data obtained are somewhat outside the applicability range of the semiempirical Eq. (16) (our data spread over the range 0.84 × 108 < Gr Pr < 1.7 × 108, while it is recommended that the relationship discussed be applied at GrPr < 108), the maximum disagreement does not exceed 14%, which is close to the experimental error. In all experiments of this series there was no systematic change in wall temperature over the test section length and perimeter. Experiments on film condensation were carried out with p = 2.0-3.7 MPa, AT = 78-105 K, and q = 0.68-0.89 M W / m 2. Superficial steam velocity during these experiments was within the range 0-0.09 m / s and did not influenced heat transfer. Figure 3 shows a comparison of the results obtained in the present work (the top cluster of points) with other available experimental data on the condensation of vapors of various substances (the bottom group of points) as well as with results of calculations using Labuntsov's formula [12]. It is seen that there is good agreement between these sets of data. The relationship [12] Ref
= 0.81Z3/4•
8O
J
o
] 6"
I 8
t /0 ~
(17)
is one of the most precise. It takes account of the variability of condensate properties, subcooling, and effects releated to liquid film wave motion.
I
60
(16)
I 2
Figure 2. Comparison of heat transfer measurements with water natural convection values calculated with Eq. (16).
732 A.A. Avdeev et al. r
I
/00 8 0 g-
2
tO ?3 6
J¢
4
I
2
I
4'
I
6
~ ~ /
I
a
I
//
I
6
I
I to
1 a2
I
4
I
6
I
I
/00
1
2
I
4/
I
G
I] ,~
Figure 3. Comparison of heat transfer measurements with film condensation values calculated with Eq. (17). ((3, data of previous works; A, data of the present work) During this series of experiments, a change in test section wall temperature over the perimeter caused by an increase in condensate film thickness was observed. However, the maximum change in the wall-liquid temperature difference over the perimeter did not exceed + 6% and did not exert any appreciable influence on heat transfer. It should be noted that the results of control experiments on natural convection and film condensation somewhat widen the range of parameters studied.
St = 0.1 (Re Fr Pr 2 ) - 025
DISCUSSION A series of experiments to measure heat transfer to a cooled cylindrical surface immersed in a saturated water layer through which steam is being bubbled were carried out in the following range of operating conditions: p = 0.72-3.8 MPa; w0 = 0-0.085 m/s; q~ = 0-0.18; q = 0.36-1.1 M W / m 2. The main results of these measurements are given in Table 1. Less than 10% of the total steam supplied to the bubbling vessel condensed on the test section surface. Therefore, in processing the data, we ignored the change in superficial steam velocity along the test section height. During experiments on bubbling heat transfer there was no systematic change in wall temperature over the test section length and perimeter. The arithmetic mean of time-averaged readings of all thermocouples recording wall temperature was used as a reference value. As steam content (superficial steam velocity) increased, the heat transfer coefficients monotonously increased (Fig. 4). On a logarithmic scale, the relationship a = f(w o) is well approximated by straight lines corresponding to power low
a = const ~n
well illustrated by Fig. 4 in which experimental points obtained at the lowest pressures (p = 0.72-0.76 MPa) and those obtained at the highest pressures (p = 3.53-3.83 MPa) are plotted. A comparison of measurement results with the known correlations that generalize experimental data obtained under atmospheric pressure shows systematic disagreement. As an example, Fig. 5 shows a comparison of our measurement results with the Deckwer correlation [6], (19)
as well as with other correlations [1, 8, 9]. It is seen that the known correlations result in a systematic underestimation of heat transfer by 30-80%. Several different explanations for these disagreements are possible. As we see it, a change in phase circulation hydrodynamics in the bubbling colum with growing pressure [13, 14], which is not taken into account by formulas like Eq. (19), is the most likely reason. PRACTICAL S I G N I F I C A N C E / U S E F U L N E S S High values of heat transfer coefficients that reach magnitudes on the order of 10-12 k W / ( m 2. K) with moderate steam content values attract our attention. These values somewhat exceed the values typical for film condensation. To maintain such considerable values of heat transfer coefficients with water, cross flow velocities on the order of 10 m / s are required. It seems to be reasonable to revise the commonly used approach to the design of built-in heat exchangers for natural circulation loops, locating the greater part of their surface below the coolant level in a bubbling layer.
(18)
where the exponent n falls within the limits of 0.25-0.29. There is a certain divergence between experimental points of different pressures in coordinates a =f(q~). This is
CONCLUSIONS An experiment was carried out to heat transfer with high-pressure steam condensation on a cooled surface
Heat Transfer in Bubble Layers i
8 8 7 6 # ,/,/
-
I
I
-~
I
733
I
/
m
¢
..
I
1
I
I
I
I
e
~
a
8
/o
ao
u2, 9;
Figure 4. Dependence of coefficient of heat transfer with bubbling on steam content.
Table 1. Measurement Results 1
2
Pressure (MPa)
Volume Steam Content (%)
0.755 0.755 0.755 0.775 0.775 0.746 0.730 1.18 1.18 1.26 1.29 1.32 2.19 2.24 2.28 2.28 2.28 2.33 2.33 2.85 2.97 2.80 2.94 3.24 2.70 3.06 2.89 2.70 2.75 2.75 3.53 3,73 3,73 3.83 3.83 3.78 3.53 3.53 3.58
2.0 3.0 4.0 5.7 7.3 8.5 10.5 7.5 8.2 9.8 11.0 12.8 5.0 6.8 9.3 10.5 12.7 13.4 13.4 3.0 3.5 3.8 7.0 7.5 8.7 9.0 13.3 14.0 14.8 15.3 7.2 10.0 13.0 14.5 16.5 18.0 15.0 17.5 18.0
3
Superficial Steam Velocity (m / s) -----0.060 0.070 ---0.060 0.070 --0.030 0.040 0.060 0.070 0.070 -0.023 0.023 0.021 0.023 0.040 -0.080 0.082 0.084 0.085 0.023 0.041 0.052 0.053 0.063 0.080 0.070 0.080 0.080
4
5
6
7
8
Inlet Water Temp. (°C)
Outlet Water Temp. (°C)
Wall Temp. (°C)
Water Flow Rate (g / s)
Heat Transfer Coeff. [kW / (me . K)]
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5,0 5,0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 24.0 24.0 24.0 24.0 20.5 5.0 20.5 20.5 20.5 20.5 24.0 24.0 24.0 24.0 24.0 24.0 5.0 5.0 5.0
36.0 52.0 58.0 62.0 63.0 64.5 66.5 61.0 63.0 65.0 66.0 68.0 59.0 62.0 64.0 67.0 55.0 57.0 58.0 64.0 71.7 69.3 78.3 61.7 68.5 60.0 76.5 75.5 76.5 78.0 61.7 65.5 67.9 70.3 70.8 71.1 48.5 54.0 51.5
114.0 119.0 122.5 124.0 125.5 127.0 128.0 128.0 130,0 131.0 133.0 133.0 134.0 143.0 143,0 143.0 144.0 145.0 146,0 130.0 133.5 132,0 137.5 139.0 134.0 133.0 141,0 140,0 142.0 142.0 139.0 140.0 145.0 148.0 148.0 149,0 144,0 146.0 146.5
42.5 26.7 26.5 26.7 26.2 25.8 25.7 35.6 36.0 38.3 38.6 37.3 46.3 46.3 45.0 44.7 59,7 58,3 58.7 37.6 59.0 59.0 58.5 81.0 61.0 55.0 61.0 61.0 61.0 61.0 81.0 81.0 81.0 81.0 81.0 81.0 82.7 81.3 81.7
7.12 7.46 9.00 9.85 10.2 11.1 12.2 9.85 10.7 11.4 11.8 11.6 8.80 10.3 10.2 10.6 11.6 11.8 12.2 6.47 8.23 7.96 9.73 8.99 9.09 8.66 11.0 11.1 11.5 11.8 8.56 9.24 10.3 11.0 11.0 11.1 10.6 12.0 11.4
734 A.A. Avdeev et al.
8~
- -
l
I
•
r
.........
] .....
q
-F--T--
I
~teiff~ l~e~spach P.N.{ g-7 0
/ - - / 4 a ~ WF'.{ ¢7 --
Q~
~o//
o,o~
o
o,o6 00
_
o,o~ A
0
0 0
°°°
-_
8o o
I
1
I
I
I
I
I
I
I
6
8
¢o
~0
ko
6o
~
¢oo
~oo
I~PeP~
Figure 5. Comparison of heat transfer measurements at bubbling (points) with calculations made using known formulas (lines).
~
immersed in a bubbling layer for the first time. The results of measurements are not described by the known semiempirical correlations constructed on the basis of data on heat transfer with gas bubbling at atmospheric pressure. To ascertain the causes of these disagreements further experimental studies are required in the pressure range 0.1-1.0 MPa.
Tout bulk cooling water temperature at test section outlet, K AT liquid-wall temperature difference, K t time, s V measuring vessel volume, m 3 wo superficial steam velocity in bubbling column, m / s Z parameter in Eq. (17) ( = Ga 1/3 h s AT/r~s), dimensionless
NOMENCLATURE
Greek Symbols a heat transfer coefficient, W / ( m 2. K) /3 temperature coefficient of liquid volume expansion [= (op#ar)JpA, K -1 • condensate properties variability correction, Eq. (17) 3 3 {[(A~/As)( IzJl~w)] 1/8 }, dimensionless A thermal conductivity of liquid, W / ( m . K) /z dynamic viscosity of liquid, N . s / m E v kinematic viscosity of liquid, m2/s p density, k g / m 3 Pout cooling water density at test section outlet, k g / m 3 lop water density at pressure taps, k g / m 3 Ptp two-phase mixture density [= pt(1 - ~) + Pv~], kg/m 3 true volumetric steam content averaged over column cross section, dimensionless
a thermal diffusivity of liquid (= A/ptCp), m / s Cp heat capacity of liquid at constant pressure, J / ( k g .
K) d c bubbling column diameter, m d b bubble diameter, m d t test section diameter, m Fr Froude number (= w~/gdb), dimensionless G cooling water flow rate, k g / s Ga Galilei number (= gH3/v~), dimensionless Gr Grashof number (= g/3ATd~/v2), dimensionless g gravitational acceleration, m / s 2 H linear scale, Eq. (17), m h vertical distance between pressure difference taps, m L t test section length, m N electric heater power, W Nu Nusselt number ( = ad/h), dimensionless Pr Prandtl number ( = v/a), dimensionless p pressure, Pa Ap hydrostatic pressure difference, Pa Q heat losses into the environment, W q heat flux density, W / m 2 Re Reynolds number ( = wodb/v), dimensionless Ref modified Reynolds film number {= qH/rlxs[1 +
(3/8)(Cp AT~r)]}, r St T Tin
specific heat of vaporization, J / k g Stanton number [(= q/( plWoCp AT)], dimensionless temperature, K bulk cooling water temperature at test section inlet, K
Subscripts
l s v w
liquid in saturation state vapor wall
REFERENCES 1. Hart, W. F., Heat Transfer in Bubble-Agitated System. A General Correlation, Ind. Eng. Chem. ProcessDes. Dev., 15, 109-114, 1976. 2. Burkel, W., Der Warmenbergang and Heiz- und Kuhlfachen in begasten Flussigkeiten, Chem. Ing. Techn.,5, 265-268, 1972. 3. Saxena, S. C., and Valdivel, R., Heat Transfer from a Tube Bundle in a Bubble Coumn, Int. Commun. Heat Mass Transfer, 15, 657-667, 1988.
H e a t Transfer in Bubble Layers 4. Tarat, E. Ya., Hoze, A. N., and Sharov, Yu. I., Investigation of Heat Transfer from a Tube Bundle in a Foam Layer, in Heat and Mass Transfer, Vol. 4, pp. 336-343, Academy of Science of BSSR, Minsk, 1968. 5. Sokolov, V. N., and Salamahin, A. D., Heat Transfer from Gas-Liquid System to a Heat Transfer Element Wall Under Bubbling Conditions, J. Appl. Chem., 35, 1022-1026, 1962. 6. Deckwer, W. D., On the Mechanism of Heat Transfer in Bubble Column Reactors, Chem. Eng. Sci., 35, 1341-1346, 1980. 7. Joshi, J. B., Sharma, M. M., and Shah, Y. T., Heat Transfer in Multiphase Contactors, Chem. Eng. Commun. 6, 257-271, 1980. 8. Kast, W., Analyse des Warmeubergangs in Blasensaulen, Int. J. Heat Mass Transfer, 5, 329-336, 1962. 9. Steiff, A., and Weinspach, P. M., Heat Transfer in Bubble Systems, German Chem. Eng., 1, 150, 1978. 10. Kutateladze, S. S., and Nakoryakov, V. E., Heat and Mass Transs-
735
fer and Waves in Gas-Liquid Systems, Nauka, Novosibirsk, 1984 (in Russian). 11. Grigoriev, V. A., and Zorin, V. M., Eds., Heat and Mass Transfer. Thermal Engineering Experiment: Handbook, Energoatomizdat, Moscow, 1982 (in Russian). 12. Labuntsov, D. A., Heat Transfer with Film Condensation of Pure Vapors on Vertical Surfaces and on Horizontal Tubes, Thermal Eng., 7, 72-79, 1957. 13. Avdeev, A. A., Hydrodynamics of Bubbling, Thermal Eng., 11, 42-46, 1983. 14. Avdeev, A. A., and Sirenko, E. I., The True Volume Steam Content with Bubbling, Thermal Eng., 11, 43-46, 1984.
Received April 1, 1991; revised February 12, 1992