Condensation heat transfer characteristic in the presence of noncondensable gas on natural convection at high pressure

Nuclear Engineering and Design 239 (2009) 688–698

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Condensation heat transfer characteristic in the presence of noncondensable gas on natural convection at high pressure Jong-Won Kim a,∗ , Yeon-Gun Lee a , Hyoung-Kyoun Ahn b , Goon-Cherl Park c a

Department of Nuclear Engineering, Seoul National University, Nuclear Thermal Hydraulics Engineering Laboratory, 31-207, San 56-1, Sillim-dong, Kwanak-gu, Seoul, 151-744, Republic of Korea Future and Challenge Technology, Sillim-dong, Kwanak-gu, Seoul, 151-744, Republic of Korea c Department of Nuclear Engineering, Seoul National University, Nuclear Thermal Hydraulics Engineering Laboratory, 32-213, San 56-1, Sillim-dong, Kwanak-gu, Seoul, 151-744, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 16 May 2008 Received in revised form 1 December 2008 Accepted 3 December 2008

a b s t r a c t The steam–gas pressurizer in integrated small reactors experiences very complicated thermal-hydraulic phenomena. Especially, the condensation heat transfer with noncondensable gas under natural convection is an important factor to evaluate the pressurizer behavior. However, few studies have investigated the condensation in the presence of noncondensable gas at high pressure. In this study, therefore, a theoretical model is proposed to estimate the condensation heat transfer at high pressure using the heat and mass transfer analogy. For the high pressure effect, the steam and nitrogen gas tables are used directly to determine the density of the gas mixture and the heat and mass transfer analogy based on mass approach is applied instead of that based on the ideal gas law. A comparison of the results from the proposed model with experimental data obtained from Seoul National University indicates that the condensation heat transfer coefficients increase with increasing system pressure and with decreasing mass fraction of the nitrogen gas. The proposed model is also compared with other conventional correlations proposed in the literature. The proposed model demonstrates the capability to predict the condensation heat transfer coefficients at high pressure better than any other correlation. Finally, the condensate rate is compared to verify the application of the heat and mass transfer analogy at high pressure. The comparison results confirm that the heat and mass transfer analogy can be applied to evaluate the condensation heat and mass transfer at high pressure. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Small- and medium-sized nuclear reactors have been constructed as new power sources with competitive power in views of economy, maintenance, and safety since their usage can be extended to seawater desalination, district heating, small power generation, and ship propulsion. The future dispersed power for producing and distributing domestic and industrial heat and electricity is emphasized. In line with this trend, small and medium reactors have been developed in many countries with advanced nuclear technology. The technical viability of using nuclear power for district heating and other industrial processes has been demonstrated both in dedicated nuclear heating plants and in heat and power cogeneration plants. Dedicated nuclear heating systems were designed and a few have been built and operated in China and Russia. The plants in China were constructed for demonstration purposes, whereas the Russian plants aimed at supplying

∗ Corresponding author. Tel.: +82 2 880 8339; fax: +82 2 889 2688. E-mail address: [email protected] (J.-W. Kim). 0029-5493/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2008.12.011

settlements in northern Russia. The Regional Energy Research Institute for Next Generation, RERI, has been developing a Regional Energy Reactor of 10 MWth (REX-10) (Kim et al., 2007). Since this regional energy reactor will be located relatively close to a residential area, highly enhanced safety features are required compared with current nuclear power plants. The REX-10 reactor system is designed based on SMART (System-integrated Modular Advanced ReacTor) (Chang, 2002) and its system pressure and capacity are determined properly for a regional energy reactor. To guarantee highly improved safety, the entire primary systems such as core, pumps, main heat exchangers (steam generators) and pressurizer are arranged in a single pressure vessel. For the incorporation of an inherent safety and passive system into the design characteristics of regional energy reactors, the natural circulation operation system is adopted. A schematic diagram of REX-10 is presented in Fig. 1. In nuclear reactors, various pressurizers are adopted according to their characteristics and usages. The additional active systems such as heater, pressurizer cooler, spray and thermal insulator are essential for a steam or gas pressurizer. Thus, REX-10 adopted the steam–gas pressurizer for the passive operation and selfpressurization. With a steam–gas pressurizer, additional systems

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689

Nomenclature cp D Dout g Gr h hfg hm k L M ˙ m m Nu P Pr q R Re Sh T w X

specific heat [J/kg K] diffusion coefficient [m2 /s] outer diameter [m] constant of gravity [m/s2 ] Grashof number heat transfer coefficient [J/kg] latent heat of vaporization [J/kg] mass transfer coefficient [J/kg] thermal conductivity [W/m K] condensing length [m] molar mass mass flow rate [kg s] interfacial mass flux [kg m2 s] Nusselt number pressure [bar] Prandtl number heat flux [W/m2 ] Universal gas constant Reynolds number Sherwood number temperature [K] mass fraction of noncondensable gas molar fraction

Greek symbols ı film thickness [m]  viscosity [Pa/s]  density [kg m3 ] Subscript avg average b bulk cond condensation conv convection f film g gas I interface in inlet l liquid max maximum mix mixture gas out outlet sat saturation tot total v vapor w wall 0 initial

are not required due to the use of steam and noncondensable gas (nitrogen gas) as pressure-buffering materials (Kim, 2005). However, there have been few investigations and reports on the steam–gas pressurizer compared with the steam pressurizer. Although the Russian Federation has developed a nuclear reactor with a steam–gas pressurizer, such as AST-500, few data are available. Although the structure of the steam–gas pressurizer is simple, the thermal-hydraulic phenomena are very complex. Especially, the effect of condensation heat transfer in the presence of noncondensable gas under a natural convection is important to evaluate the pressurizer behavior. The steam generated from primary coolant is mixed with nitrogen gas which is pre-filled to control the pressure. Film condensate is formed on the cold wall of the steam–gas pressurizer and falls down to the coolant, as shown

Fig. 1. Schematic diagram of REX-10 (Regional Energy rX – 10 MWth ).

in Fig. 2. Thus, the condensation heat transfer is the dominant heat transfer mechanism to ensure the pressure sustainability in the steam–gas pressurizer. The presence of noncondensable gas has been reported to decrease substantially the condensation heat transfer (Minkowycz and Sparrow, 1966). Most previous studies mainly focused on steam/air condensation inside the containment structure. These studies covered the system pressure up to 1.0 MPa. However, since the system pressure in the REX-10 steam-gas pressurizer is 2.0 MPa, the heat transfer rates at high pressure condition should

Fig. 2. Heat transfer mechanism in the steam–gas pressurizer.

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be obtained to support the analysis of the steam–gas pressurizer behavior. Several researchers have performed experiments and investigated the effect of system pressure on natural convection condition. Gerstmann and Griffith (1967) found that the heat transfer rates increase with increasing system pressure because the density of the gas components increases with an increase of pressure. The earlier measurements used for the design and equipment qualification of containments were obtained by Uchida et al. (1965) and Tagami (1965). Uchida performed his experiments under natural convection condition. A vertical plate of dimensions 140 mm by 300 mm was installed in a closed vessel filled with steam and noncondensable gas. The air–steam mixture experiment was tested in the range from 0.1 to 0.28 MPa. Their results indicated that the condensation heat transfer coefficients decreased as a function of the mass fraction of noncondensable gas. Dehbi (1991) tested the condensation heat transfer on the cooling pipe in the closed cylinder chamber. On the surface wall of the cooling pipe, the condensation film was formed at a pressure range about 0.15–0.45 MPa. The heat transfer coefficients decreased with increasing air gas mass fraction and decreasing of total pressure. It was concluded that the total pressure and mass fraction of noncondensable gas were dominant factors to determine condensation heat transfer coefficients. Falkov et al. (2004) at OKBM carried out an experiment on steam condensation in large volume at high pressure. Experimental data of steam condensation on the vertical wall were obtained within a pressure range from 0.4 to 3.0 MPa and a mass fraction of noncondensable gas from 0.15 to 0.8. However, few data are available. Thus, Ahn et al. (2007) in Seoul National Univ. performed experiments to study the condensation heat transfer with noncondensable gas on natural convection at high pressure. The cylindrical chamber with a diameter of 300 mm, height of 1500 mm was filled with the water and the mixture of steam and nitrogen gas on natural circulation. A 650 mm-long condensing tube was also set up at the center of the autoclave chamber to condense the steam on the cold wall. From this experimental study, the condensation heat transfer coefficient at high pressure could be obtained. In addition, the theoretical investigations have been performed. In approaching the condensation heat transfer, the following viewpoints are very important. One is condensation model to calculate the total heat transfer coefficient. The other is heat and mass transfer analogy to relate the condensation heat transfer to the generation of condensate. From the viewpoint of condensation model, the modeling procedure has been already standardized since Rohsenow and Choi (1961) proposed condensation model. Since then, Corradini (1984), Kim and Corradini (1990), Peterson et al. (1993) and Herranz et al. (1998) extended and completed this model. The fundamentals of the condensation model and applied heat transfer correlations are presented in more detail in the next section. In the heat and mass transfer analogy, Ambrosini et al. (2006) summarized theoretical bases. It was reported that it was possible to consider the variety of forms that are presently used to express the heat and mass transfer analogy, basing on different formula and expressions. For example, one is the mass approach adopted by Spalding (1960) and Lienhard (1987) and the other side is the molar approach adopted by Bird et al. (1960). Especially, the approach by Peterson et al. (1993) and Peterson (1996) is derived from the classical molar approach with the assumption of the ideal gas behavior of the involved gas and additional approximations for evaluating steam properties. Since each approach is based on different assumption, it is important to select the adequate heat and mass transfer analogy for a given application. In this study, a theoretical model is proposed to estimate the condensation heat transfer with noncondensable gas on natural convection at high pressure using the heat and mass transfer analogy. The proper heat and mass transfer analogy to high pressure

condition is found and this proposed model is compared with the experimental data from Seoul National University. In addition, this model is verified by comparison with OKBM experimental data and compared with other conventional correlations and models. Moreover, the condensate rate is also compared with the experimental and calculated data to verify the propriety of the heat and mass transfer analogy at high pressure. 2. Fundamentals of the model The condensation heat transfer model based on the diffusion layer model with the heat and mass transfer analogy is used to describe the wall condensation phenomenon. As shown in Fig. 3, a diffusion layer is formed on the mass transfer surface during the steam condensation process from the steam–gas mixture. The partial pressure of the noncondensable gas increases near the condensate film and consequently decreases the condensation heat transfer rate. At first, it is assumed that noncondensable gas is in mechanical and thermal equilibrium with the vapor. If the total pressure and mass fraction of noncondensable gas are given, the temperature of the gas mixture and partial pressures of the vapor and noncondensable gas can be determined. Moreover, the wall temperature can be determined as the wall subcooling between the gas mixture and wall is given. The condensation heat transfer model is derived from the heat balance at the liquid/gas interface (Faghri and Zhang, 2006). The heat transfer rate from the condensate film to the wall is the same as that through the gas mixture boundary layer. The heat transfer through the gas mixture boundary layer consists of two different heat transfer mechanisms: the convective heat transfer in the diffusion layer and the condensation heat transfer carried by mass transfer from the vapor to the interface (Herranz et al., 1998). Therefore, the following equation can be obtained: hf (Ti − Tw ) = (hcond + hconv )(Tmix − Ti )

(1)

Fig. 3. Condensation heat transfer characteristics with noncondensable gas on the natural convection system.

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By applying the thermal-resistance concept, the total heat transfer coefficient can be derived as htot =

1 1/hf + 1/hcond + hconv

(2)

2.1. Condensate film In general, the heat transfer coefficient on a vertical surface in a condensate film region is given by the Nusselt equation (Nusselt, 1916). The film thickness, ı(L), is expressed as a function of the vertical length, L. Then, the heat transfer coefficient through the condensate film can be obtained through Eqs. (3) and (4):

 ı(L) =

hf (L) =

4kl3 l (Tsat − Tw )L

1/4

gl (l − v )hfg

(3)

(4)

where hfg is a modified latent heat which accounts for the condensate subcooling and the temperature jump across the film. In addition, Kutateladze’s correlation (Kutateladze and Gongonin, 1979) multiplied hf (L) by  for the enhancement in heat transfer by accounting for film waviness and the rippling effect as given Eq. (5).  =

Re0.04 l

(5)

2.2. Gas mixture The heat and mass transfer analogy, a well known tool for the prediction of heat and mass transfer involving laminar or turbulent diffusion, is used to analyze the gas mixture region. By applying the heat and mass transfer analogy, which replaces the Nusselt and Prandtl numbers with the Sherwood and Schmidt numbers, respectively, the natural convective and condensation heat transfer coefficient can be obtained simultaneously. The Nusselt number for natural convection is obtained by the natural convective heat transfer coefficient as a function of the Grashof number and the Prandtl number in Eq. (6) (Rohsenow and Choi, 1961). The laminar, transition and turbulent flow regions are considered separately, since the natural convection flow is strongly dependent on the length of the vertical wall: Nu = 0.56(Gr Pr)1/4 Nu = 0.13(Gr Pr)1/3 Nu = 0.021(Gr Pr)2/5

for laminar flow (104 < Gr Pr < 108 ) for transition flow (108 < Gr Pr < 1010 ) for turbulent flow (1010 < Gr Pr) (6)

The condensation heat transfer coefficient can be defined as follows: hcond =

mcond hfg

(7)

Tmix − Ti

In order to determine mcond , the mass balance at the interface is considered as (Siddique, 1992):



mcond = −D

∂wv ∂y



+ wv,i (mtot )i

mcond =

wv,b − wv,i [−D(∂wv /∂y)]i = hm , 1 − wv,i 1 − wv,i

Sh D wnc,i − wnc,b L wnc,i

(10)

The Sherwood number for mass transfer, which is analogous to the Nusselt number for heat transfer, is derived from the Nusselt number using the heat and mass transfer analogy. In the same way, the Prandtl number is replaced with the Schmidt number. The boundary layer thickness is reduced due to the apparent suction effect caused by the mass of steam towards the interface. The suction effect due to condensation in the diffusion layer is considered. The Sherwood number is multiplied by the following correction factor, , to take into account the suction effect (Herranz et al., 1998): Xnc,i Xnc, avg

(11)

The interfacial mass flux, mcond , is calculated by the modified Sherwood number from Eq. (10). Therefore, the condensation heat transfer coefficient can be obtained using Eq. (7). 2.3. Determination of the interface temperature on the condensation surface When the total system pressure and the mass fraction of noncondensable gas are given, the steam partial pressure of the bulk gas mixture can be determined. As the temperature difference between the gas mixture and wall is known, the temperature of the gas mixture and wall are obtained according to the steam partial pressure. However, the interface temperature and the mass fraction at the interface are unknown and must be determined analytically for accurate analysis. In some research (Arcipreti, 2006), when modeling the film condensation phenomenon in a CFD code, the condensate film layer is neglected as it is much thinner than the boundary layer. Moreover, when the mass fraction is high, the condensation heat transfer is more dominant than the film heat transfer, so that the film heat transfer can be negligible. Thus, the temperature difference in the film is assumed to be small and the interface temperature is assumed to be equal to the wall temperature. However, the interface temperature should be calculated for accurate analysis. In the present calculation of the film, natural convective and condensation heat transfer coefficients, the interface temperature is assumed to be Tw + Tmix /2. The updated interface temperature can be obtained by the following equation which is rewritten from Eq. (1): Ti∗ =

hf Tw + (hcond + hconv )Tmix hf + hcond + hconv

(12)

The interface temperature can be determined thought this iteration procedure. Then, the updated film, natural convective and condensation heat transfer coefficients are obtained from the updated interface temperature. This interface temperature is considered to be fully converged when the maximum temperature change from iteration to iteration is smaller than a prescribed value, 10−7 . Fig. 4 shows the calculation procedures. Finally, the total heat transfer coefficient is determined by the heat transfer coefficients of the condensate film and gas mixture.

(8)

i

2.4. Thermal properties at high pressure

As the condensate surface is impermeable to the noncondensable gas, which means that (mtot )i is mcond , Eq. (8) can be simplified to mcond =

where hm is the mass transfer coefficient. Using the definition of the Sherwood number, Eq. (9) can be rewritten as

=

kl ı(L)

691

(9)

Previous studies mainly investigated steam/air condensation in containment at system pressures to 1.0 MPa. In applying the heat and mass transfer analogy, the molar approach uses the molar concentration term which is expressed using the ideal gas law (Kim and Corradini, 1990; Peterson et al., 1993; Herranz et al., 1998; Bird et

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Fig. 4. Calculation procedures.

al., 1960). However, steam and noncondensable gas undergo significant interaction at high pressure, resulting in deviation of the gas thermal properties from the ideal gas law (Anderson, 1994). The compressibility is 0.9852 at 0.1 MPa and 0.8865 at 2.0 MPa which is the operation pressure of REX-10. Thus, the ideal gas law underestimates the density of steam and noncondensable gas. Especially, since the Grashof number which governs the natural convection is also proportional to the density of the gas mixture, it is important to obtain the accurate density of gas mixture. Thus, the heat and mass transfer analogy based on mass approach is applied to resolve discrepancy of molar one based on the ideal gas law. In addition, in order to determine the density of gas mixture, the nitrogen gas property tables, P-PROPATH-N2 and P-PROPATH-H2 O (DME, 2001) or virial coefficients are used. For diffusion coefficients of binary gas mixtures, the following relation is used with account for the diffusion coefficient as a function of pressure and temperature (Falkov et al., 2004): Di = D0,i

P0 P

 T n  T0

1 − 0.3

P 200



(13)

where D0,i is the diffusion coefficient of the i-th gas with respect to steam at T = T0 and P = P0 . P0 and T0 are the ambient pressure and temperature of 1.013 bar and 273 K, respectively. The exponent, n, is 1.75 and the value of nitrogen gas diffusion coefficient in steam, D0 , is 0.24 × 10−4 [m2 /s]. The correction term (1 − 0.3(P/200)), for gas mixture imperfection under the increased pressure is also introduced. 3. Condensation heat transfer experiment at high pressure with nitrogen gas Seoul National University performed an experimental study on condensation heat transfer with noncondensable gas on natural convection at high pressure. The experimental facility consisted of a test chamber, chiller and data acquisition system. The chamber design of the condensation heat transfer experiment facility at high pressure is shown in Fig. 5. To estimate the condensation heat transfer at high pressure, an autoclave chamber was used to endure the high pressure condition and a vertical cooling pipe was also set up at the center of the autoclave chamber to condense the steam on the cold wall. The water coolant flowed from the chiller into the cooling pipe and formed the condensate film on the outer wall

Fig. 5. Schematic diagram of the test chamber in the Seoul National University experiment.

of the cooling pipe. The diameter of the cooling pipe was 0.04 m and the length of cooling pipe was 0.65 m. The horizontal tube sections inside the chamber were also insulated to prevent additional heat transfer out of effective condensing region. At the bottom of chamber, an electric heater was installed to evaporate the water and produce a mixture of steam and nitrogen gas. The maximum heater power was 20 kW and controlled by the heater controller. In addition, the test chamber was insulated by asbestos to reduce the heat loss to the atmosphere. Total 19 K-type thermocouples were installed in order to measure temperature of mixture gas (12), outer wall of cooling pipe (5), inlet and outlet coolant (2). The thermal stratification in the test chamber was estimated according to the temperature of mixture gas along the vertical level and the mass fraction of noncondensable gas was obtained by measuring the temperature of mixture gas. In addition, the condensation heat transfer coefficient could be obtained based on a coolant energy balance by measuring the inlet and outlet temperatures of the coolant. In particular, the outside of the thermocouple was insulated with silicon to prevent inaccurate measurements caused by direct contact with the condensate film. Shielded thermocouples were installed because the mechanical damage should be considered in the high temperature and pressure condition. To measure the condensate water, a condensate gauge was installed out of the autoclave chamber. At the end of the test section of the condensing pipe, a plate to gather condensing water was installed and linked with the condensate gauge through a small tube. The total pressure was measured at the top of the test chamber by a pressure gauge. Nitrogen gas controlled by a mass flow regulator was injected into the autoclave chamber through the gas injection valve to set the total pressure. The air which resides in the autoclave chamber has to be removed to inject the required amount of nitrogen gas. To achieve such a condition, the water at the bottom of the chamber was heated

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Fig. 6. (a) Condensation heat transfer coefficients (0.4 MPa); (b) condensation heat transfer coefficients (0.7 MPa); (c) condensation heat transfer coefficients (1.2 MPa); (d) condensation heat transfer coefficients (1.5 MPa); (e) condensation heat transfer coefficients (2.0 MPa).

first so that generated steam pressurized the chamber. When the total pressure reaches about 0.2–0.3 MPa, the ventilation valve opens to discharge air and steam mixture until the pressure equals to the saturation pressure corresponding to the mixture gas temperature. After degassing, the nitrogen gas is injected to meet the test condition, considering thermal expansion of the mixture gas. The steam generated by evaporation increases the system pressure to the required test pressure. A steady state was established controlling the heater power or changing the mass flow rate of the

coolant from the chiller. At steady state, the total pressure, the supplied power, the removal heat flux through the cooling pipe and the temperatures of the wall, mixture gas and coolant were measured at 2-min intervals. As the tenable temperature inside the cooling pipe cannot be measured, it is inevitable to obtain the average condensation heat transfer coefficient over the length of the cooling pipe. The average condensation heat transfer coefficients could be obtained using the coolant energy balance, as shown in Eq. (14). At the end of each run, the valve between the cham-

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ber and the condensate gauge opens to measure the condensate rate. htot =

˙ p (Tout − Tin ) mc Dout L(Tmix − Tw )

(14)

The experimental data from Seoul National University, shown in Fig. 6(a)–(e), indicated that the condensation heat transfer coefficients increased with decreasing mass fraction of nitrogen gas or increasing total pressure. The experimental results are presented in more detail in the next section. 4. Results and discussions 4.1. Condensation heat transfer coefficient at high pressure To assess the feasibility of the proposed model at high pressure condition, the proposed model is compared with the experimental results from Seoul National University and OKBM. The existing correlation and model for the condensation heat transfer coefficient are compared with those from the present proposed model. The experiments were carried out with the pressure varying from 0.1 MPa to 2.0 MPa to determine the pressure effect on the condensation heat transfer. The mass fraction of the nitrogen gas in the experiment was varied over the range from 0.05 to 0.7, considering the mass fraction of the nitrogen gas of the REX-10 steam–gas pressurizer. The calculation is conducted with a condensing length of 0.65 m while the length governing the natural convection was 1.05 m. The average condensation heat transfer coefficients over condensing section are obtained from temperature data and mass flow rate. In general, the stratification in mixture gas is important to estimate condensation heat transfer coefficient. The stratification may be expected when two gases of different densities are mixed. In order to evaluate the stratification of the steam and nitrogen gas, the temperature of the gas mixture were measured at 50, 200, 350, 500 and 650 mm from the top of chamber. As a result, there was no difference in the temperature of the mixture gas along the vertical level. That is, the mass fraction of noncondensable gas was almost same along the vertical level. Therefore, it was concluded that stratification did not occur in the autoclave chamber and the gas mixture was mixed well due to natural convection. The outer wall temperatures of cooling pipe at same level were measured with gas mixture. As a result, the difference in temperature between the top and bottom of outer wall was within 5.0 ◦ C. Thus, there is no problem in obtaining the average condensation heat transfer coefficient. The uncertainty analysis was performed to evaluate the inherent experimental error originated from the instrumentation, data acquisition and so on. The condensation heat transfer is the function of five variables from Eq. (15) and then given as ˙ Tout , Tin , Tmix , Tw ) h = h(m,

Fig. 7. Comparison of the experimental and predicted data.

increased with decreasing mass fraction of nitrogen gas or increasing the total pressure. From the viewpoint of the wall subcooling, which is another dominant factor governing the condensation heat transfer, the inlet temperature of the coolant from the chiller was almost constant during the single-pass cooling in each case. However, it is hard to keep wall subcooling to be constant, since the experimental conditions in the test chamber were different in each test. Thus, the model results are given in Fig. 6 with the most probable wall subcooling on each pressure condition from the experiment whereas the experimental data have various wall subcoolings. The model results shown in these figures exhibited good agreement with the experimental data by Seoul National University. Fig. 7 shows the performance of the model by representing the experimental and predicted data for the total number of experimental points. Although the deviation between the experimental and calculated data in Figs. 6 and 7 is relatively large at high pressure, most of the predicted data lie within a ±20% error band and the average error is about 15%. In Seoul National University experimental study, the condensate rate was measured in some cases. To measure the condensate water, a condensate gauge was installed out of the autoclave chamber. At the end of the test section of the condensing pipe, a plate to gather condensing water was installed and linked with a condensate gauge through a small tube. The valve between the test chamber and the condensate gauge was opened for 30 s to measure the amount of condensing water. By measuring the water level, the condensate rate was calculated. Fig. 8 compares the condensate rates from the experiment and the calculation; it is evident that the model forms a reasonable prediction of the condensate rate. Consequently, the heat and mass transfer analogy can be applied to the condensation heat and mass transfer at high pressure. In the OKBM experiment, steam condensation from the steam–gas mixture in large vessels was studied on the guard vessel

(15)

Then, the uncertainty of the condensation heat transfer coefficient is estimated using the instrumental errors and experimental data, as follows:



∂ ∂h



 =

 2 m ˙

∂h ˙ ∂m

2

 + T2 mix

∂h ∂Tmix

2

 + T2w

∂h ∂Tw

2

 + T2 in

The temperature was measured by K-type thermocouples which were calibrated to have error within ±1.0 ◦ C. The maximum error of turbine flow meter is 5.0% of mass flow rate. Consequently, with 95% of confidence level, the uncertainty of condensation heat transfer coefficient was estimated to be within 25%. Fig. 6(a)–(e) show the comparison of the condensation heat transfer coefficient between the experimental and analysis data at each pressure. The condensation heat transfer coefficients

∂h ∂Tin

2

 + T2 out

∂h ∂Tout

2 (16)

model of the KMR-2 facility with a wide range of steam–gas mixture parameters. The KMR-2 experimental facility was constructed in TsKTI on the order of OKBM to simulate the thermo-hydraulics of the AST-500 primary circuit in accidents with primary circuit depressurization and heat removal failures. Fig. 9 compares the calculation results of the heat transfer coefficient during steam condensation using the proposed model based on the analogy

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Fig. 10. Effect of pressure on the condensation heat transfer coefficient. Fig. 8. Comparison of condensate rates from the experiment and proposed model.

of heat and mass exchange processes under the conditions of steam–gas mixture natural convection. This calculation used the OKBM experimental data on heat transfer coefficients during steam condensation from the steam–air mixture obtained on the KMR-2 test facility guard vessel model. The comparison is presented in the form of the dependence of heat transfer coefficients on the mass fraction of the noncondensable gas. The results showed good agreement with the OKBM experimental data, and supported the conclusion that this model accurately predicts the condensation heat transfer at high pressure. In order to verify the proposed model are compared with the existing correlations. Among the existing correlations, those of Uchida and Tagami, which are widely used in nuclear reactor design and obtained on natural convection, are shown as follows:

 W −0.7

h = 380

1−W

h = 11.4 + 284

1 − W  W

(17) (18)

Uchida and Tagami’s data have been correlated in terms of a single variable: the mass fraction of noncondensable gas. Thus, these correlations cannot estimate the effect of pressure variation. In addition, the Dehbi correlation shown Eq. (19), which is dependent on the system pressure, mass fraction of noncondensable gas,

Fig. 9. Comparison of results from the OKBM experiment and proposed model.

condensate length and wall subcooling, is also compared. h=

L0.05 [(3.7 + 28.7P) − (2438 + 458.3P) log W ] (Tmix − Tw )0.25

(19)

where 0.3 m < L < 3.5 m; 0.15 MPa < 0.45 MPa; 10◦ C < Tmix − Tw < 50◦ C. The results of the comparison between the proposed model and the conventional correlations are analyzed parametrically. Fig. 10 shows the heat transfer coefficients from the proposed model and the Dehbi correlation versus total pressure. Since the Uchida and Tagami correlations do not included a pressure term, the proposed model is compared with the Dehbi correlation. When the mass fraction, subcooling and condensing length are set at 0.3, 15 K and 0.65 m, respectively, the heat transfer coefficient from the model increases up to an operation pressure of REX-10 and the Dehbi correlation overestimates the condensation heat transfer coefficient. While the results from the Dehbi correlation exhibit a linear relation with the pressure, those from the model are similar to the saturation temperature curve according to the increase in the pressure. In general, the condensation heat transfer coefficients with pure steam are independent of the total pressure (Nusselt, 1916), whereas those with noncondensable gas are dependent on the total pressure. Therefore, the increment in the condensation heat transfer coefficient is gradually decreased with increasing pressure. Consequently, the model predicts the condensation heat transfer at high pressure better than the conventional correlations do. Fig. 11 shows the heat transfer coefficients from the proposed model according to the mass fraction of noncondensable gas. The heat transfer coefficient decreases with increasing mass fraction. In general, the condensation heat and mass transfer is significantly influenced by the presence of the noncondensable gas, even if a small quantity of noncondensable gas is included. For example, the condensation heat transfer can be reduced by up to 50% or even more when the mass fraction is as low as 0.005 (Minkowycz and Sparrow, 1966). In this model, the region where the mass fraction is very low cannot be described because the partial pressure of nitrogen gas can be obtained in the pressure range from 12.53 × 103 to 3.4 × 106 Pa. The operation pressure of REX-10 is 2.0 MPa and the mass fraction is expected to range from 0.1 to 0.4. Therefore, it is not necessary to predict the condensation heat transfer at the very low mass fraction region. Fig. 12 shows the comparison results between the conventional correlations, such as Uchida, Tagami and Dehbi, and the proposed model. The model demonstrates a similar tendency to that of the other correlations. Moreover, the model results show a better agree-

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Fig. 11. Effect of the mass fraction of noncondensable gas on the condensation heat transfer coefficient.

ment with those from the Dehbi correlation, which considers the pressure effect, than with those from the Uchida and Tagami correlations. It is concluded that the proposed model can be applied at the low pressure region. 4.2. Wavy effect on condensation heat transfer at high pressure The Nusselt’s theory assumed that the flow of condensate in the film is laminar. However, in practice the condensate surface flowing down vertical surface forms considerable waviness. This waviness causes the change in the flow distribution of the condensate flow and the decrease in the average thickness of the liquid film. Thus, the heat transfer coefficient of the liquid film increases. Especially, as the pressure increases, the natural convection in gas mixture region is more activated, which leads to sweep the liquid layer and reduce the thickness of the liquid film. Thus, the wavy effect is more considerable at high pressure as shown in Fig. 13. Fig. 13 shows the condensation heat transfer coefficients with and without the wavy effect at 0.1 MPa to 2.0 MPa. As the total pressure increases, the wavy effect on the condensation heat transfer coefficient is more outstanding. In case that the wavy effect was not considered, the film coefficients under turbulent condition at high pressure were underestimated to be about ∼10,000 (W/m2 K). As the pressure increases, the density of the

Fig. 12. Comparison of the results from the conventional correlations and proposed model.

Fig. 13. Comparison of condensation heat transfer coefficients with or without considering the wavy effect.

gas mixture increases and thus results in an increase of the heat transfer coefficient in the gas mixture. This causes the underestimation of the total heat transfer coefficient. Therefore, considering the wavy effect is significant to predict the accurate condensation heat transfer at high pressure. 4.3. Heat and mass transfer analogy at high pressure In addition, the model developed from Peterson et al. (1993) was also compared with the proposed model. In their research, the diffusion layer model introduces the Clausius–Clapeyron approximation for the difference between the vapor saturation partial pressures at the bulk and interface temperatures and the density of the condensing vapor is approximated using the ideal gas law to derive the effective condensation heat transfer coefficient. Consequently, the condensation thermal conductivity from Peterson includes the form of the ideal gas law as shown Eq. (20). kcond =

1/ Ptot M 2 Dh2fg 3 R2 Tavg

(20)

where is the noncondensable gas/vapor log mean concentration ratio given by Xg,avg /Xv,avg = − ln[(1 − Xg,b )/(1 − Xg,i )]/ ln[Xg,b /Xg,i ]. Fig. 14(a)–(c) shows the results from Peterson’s model and proposed model at 0.4, 1.2 and 2.0 MPa. As shown in these figures, there is little difference between the results from the proposed model and Peterson’s one at low pressure (0.4 MPa). Because the compressibility is near by one, the density of gas mixture from the ideal gas law is almost same with real one. On the other hand, the results at high pressure (1.2 MPa, 2.0 MPa) show the difference in the heat transfer coefficients. At high mass fraction region, the calculated results are almost same due to low mass transfer. However, as the pressure and mass fraction of noncondensable gas at the steady state of REX-10 steam–gas pressurizer operation are 2.0 MPa and 0.15, respectively, these results are more important to estimate the behavior of steam–gas pressurizer. Especially, the analysis result from the proposed model is about 23% larger than that from Peterson when the total pressure is 2.0 MPa and the mass fraction of noncondensable gas is 0.15. In addition, it was reported that the ratio of the Sherwood numbers obtained by the molar approach to the corresponding values provided by the mass approach is always greater than unity. If Sherwood number is derived in Eq. (10) and mass flux is assumed to be constant, the Sherwood number is in inverse proportion to the density of gas mixture density. That means the

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Fig. 15. Comparison of the condensation heat transfer coefficients using P-PROPATH and the ideal gas law.

transfer model with the heat and mass transfer analogy based on the mass approach can be applied more properly. Due to the error of the ideal gas law, either P-PROPATH-N2 and P-PROPATH-H2 O, or virial coefficients, are used to determine the density of the gas mixture. In reality, the ideal gas law underestimates the gas density at high pressure, which affects the heat and mass transfer in the gas mixture region because the Grashof number is a function of the gas mixture density. Thus, it is necessary to compare the results using P-PROPATH or the ideal gas law. Fig. 15 shows the comparison of the condensation heat transfer coefficients using P-PROPATH and the ideal gas law. A considerable difference is evident at high pressure and at a low mass fraction of noncondensable gas, and in the latter case the steam partial pressure is relatively large. In general, the thermo-dynamic characteristics of a monatomic gas are known to be closer to those of the ideal gas than those of a diatomic gas. Therefore, the ideal gas law cannot predict the density of a gas mixture at high pressure and at a low mass fraction of noncondensable gas. 5. Conclusions

Fig. 14. (a) Comparison of the results from the Peterson’s model and proposed model (0.4 MPa); (b) comparison of the results from the Peterson’s model and proposed model (1.2 MPa); (c) comparison of the results from the Peterson’s model and proposed model (2.0 MPa).

molar approach using the ideal gas law underestimates the density of gas mixture density. Moreover, as the total pressure increases, the ratio of the Sherwood numbers from Peterson and Lienhard also increases. Ambrosini reported that it is hard to say which one of the approaches can be considered the most coherent with experiments, as the influence of the modeling choices is likely to be masked by other effects, including experimental uncertainties. Thus, the adequate approach can be needed at high pressure condition. Therefore, it is concluded that the condensation heat

In this study, the theoretical research based on the diffusion layer model and using the heat and mass transfer analogy is applied to investigate the condensation heat transfer characteristics on a natural convection system with noncondensable gas at high pressure. In order to describe the high pressure effect, the heat and mass transfer analogy based on the mass approach is used. In addition, the property tables of steam and nitrogen gas, P-PROPATH-N2 and P-PROPATH-H2 O are used instead of applying the ideal gas law, because application of the ideal gas law leads to underestimation of the condensation heat transfer due to the lower density of the gas mixture. The analysis results show good agreement with other correlations and experimental data. The condensation heat transfer coefficient increases with increasing the total pressure and with decreasing mass fraction of the noncondensable gas. In particular, the proposed model estimates the condensation heat transfer at high pressure more accurately than the existing correlations and model. In addition, the estimated condensation rates show good agreement with those from Seoul National University. Therefore, these study results support the application of the proposed model using the heat and mass transfer analogy based on the mass approach for analyzing the condensation heat transfer with noncondensable gas at high pressure. Moreover, it is expected that the proposed model will be used to estimate the thermo-dynamic behavior of the steam–gas pressurizer.

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