Research on the steam-gas pressurizer modeled with Relap5 code

Research on the steam-gas pressurizer modeled with Relap5 code 1 corresp first-author Xi-Zhen Ma cnst_xizhen@163.com Xi-Zhen Ma cnst_xizhen@163.com cnst_xizhen@163.com 1 1 Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China Steam-gas pressurizers are self-pressurizing, and since steam and noncondensable gas are used to sustain their pressure, they experience very complicated thermal-hydraulic phenomena owing to the presence of the latter. A steam-gas pressurizer model was developed using Relap5 code to investigate such a pressurizer’s thermal-hydraulic characteristics. The important thermal-hydraulic processes occurring in the pressurizer model include bulk flashing, rainout, wall condensation with noncondensable gas, and interfacial heat and mass transfer. The pressurizer model was verified using results from insurge experiments performed at the Massachusetts Institute of Technology. It was found that noncondensable gas was one of the important factors governing the pressure response, and the accuracy of the developed model would change with different mass fractions and types of noncondensable gas. Relap5 code Noncondensable gas Heat and mass transfer Steam–gas pressurizer Condensation 1 INTRODUCTION 1 Small and medium nuclear reactors with low capital costs, high performance, and enhanced safety are widely used in district heating, seawater desalination, and ship propulsion, and are usually developed as integral reactors. Since steam-gas pressurizers are self-pressurizing and can minimize the use of active devices, such as heaters and sprayers, they are widely applied in such reactors, e.g., NHR-II (Nuclear Heating Reactor-II) [ 1 1 ], REX-10 (Regional Energy Reactor; 10MW th ), SMART (System-Integrated Modular Advanced Reactor) [ 2 2 ], etc. The thermal-hydraulic characteristics of a steam-gas pressurizer are considered the most important factors that affect the safe operation of small and medium reactors. However, studies on such pressurizers have been rarely done since reactors with a steam-gas pressurizer have not been commercialized, and the available information on such research is extremely restricted. The analysis code GARRIC was developed by Russian researchers to investigate the characteristics of steam-gas pressurizers [ 3 3 ] and the distribution of noncondensable gas, and some experiments have been conducted to verify GARRIC. Unfortunately, the results of the experiments are not available. In order to investigate the effect of a noncondensable gas on a steam-gas pressurizer, experiments were performed at the Massachusetts Institute of Technology [ 4 4 ] in which changes of pressure in the pressurizer were observed for different mass fractions of and types of noncondensable gas. Kim and Griffith [ 5 5 ] noted that condensation heat transfer at the wall played an important role in determining the pressure transient in a steam-gas pressurizer, and that the presence of even a small amount of noncondensable gas would lead to a significant reduction in heat transfer during condensation. Murase et al proposed that the condensation heat-transfer coefficient was a function of the steam/gas ratio [ 6 6 ]. A theoretical pressurizer model was developed by Kim et al . [ 7 7 ] and used in RETRAN-3D/TNT code. In this study, a steam-gas pressurizer model was developed to investigate the thermal-hydraulic characteristics of a steam-gas pressurizer using Relap5 code. Apart from the general conservation equations [ 8 8 - 10 10 8-10 ], the model includes an interfacial heat- and mass-transfer model and a wall film condensation model with the presence of noncondensable gas. The noncondensable gas plays a key role in the thermal-hydraulic processes occurring in the pressurizer since different types of and mass fractions of noncondensable gas account for significant differences in the pressure transients in the steam-gas pressurizer. The results of this study have been verified with those from pressurizer insurge experiments performed at Massachusetts Institute of Technology (MIT). 2 PRESSURIZER PHYSICAL MODEL 1 In this study, a steam-gas pressurizer’s volume is divided into three regions based on phase conditions: a steam/gas region, a steam/liquid interfacial region, and a liquid region [ 11 11 ], as shown in Fig. 1. Fig. 1. Region 1 is the gas-mixture region including vapor, noncondensable gas, and liquid droplets. Region 2 is the steam and liquid interfacial region in which the processes of heat and mass transfer occur. Region 3 is the saturated liquid region containing rising bubbles. The governing equations of the model include the phase-continuity, momentum-conservation, and energy-conservation equations [ 12 12 ]. The presence of noncondensable gas is the most important phenomenon in the pressurizer, and it is thus necessary to consider its effect when solving these governing equations. The following assumptions are made in the model: (1) Vapor and noncondensable gas in the steam/gas region are fully mixed and share the same temperature. (2) Liquid droplets in the steam/gas region are saturated before reaching the liquid. (3) Each region can have different temperatures and the three regions have the same pressure. (4) Each region has the same enthalpy at each time step. (5) The total volume of the steam-gas pressurizer is constant. The mass-conservation equations are written as <math display="block" id="M1"><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><msub><mi>v</mi><mtext>g</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>Γ</mi><mtext>g</mtext></msub><mo>,</mo></mrow></math> <math display="block" id="M2"><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><msub><mi>v</mi><mtext>f</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>Γ</mi><mtext>f</mtext></msub><mo>,</mo></mrow></math> <math display="block" id="M3"><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><msub><mi>X</mi><mtext>n</mtext></msub></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><msub><mi>v</mi><mtext>g</mtext></msub><msub><mi>X</mi><mtext>n</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>0.</mn></mrow></math> The mass-governing equations include gas-phase, liquid-phase, and noncondensable-gas-phase equations. X n is defined as the mass fraction of noncondensable gas based on the mass of the gas-mixture phase, and it is assumed that all properties of the gas phase are mixture properties of the steam/noncondensable gas mixture, where <math display="block" id="M4"><mrow><msub><mi>X</mi><mtext>n</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>M</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>M</mi><mtext>n</mtext></msub><mo>+</mo><msub><mi>M</mi><mtext>s</mtext></msub></mrow></mfrac><mo>.</mo></mrow></math> Since the noncondensable gas is assumed to be in mechanical and thermal equilibrium with the vapor phase in the model, the gas phase shares the same velocity and temperature; that is, v n = v g , and T n = T g . Without considering the mass dissipation, the total mass of the gas and liquid phases is constant, so the liquid-generation term is the negative of the vapor generation term; that is, Г g =−Г f . It is further assumed that, in the interfacial mass-transfer model 13], the total mass transfer (Г g ) can be partitioned into the mass transfer at the steam/liquid interface in the bulk fluid (Г ig ) and the mass transfer at the steam/liquid interface in the boundary layer near the walls (Г w ); that is, <math display="block" id="M5"><mrow><msub><mi>Γ</mi><mtext>g</mtext></msub><mo>=</mo><msub><mi>Γ</mi><mrow><mtext>ig</mtext></mrow></msub><mo>+</mo><msub><mi>Γ</mi><mtext>w</mtext></msub><mo>.</mo></mrow></math> The mass-transfer term Г w is evaluated as <math display="block" id="M6"><mrow><msub><mi>Γ</mi><mtext>w</mtext></msub><mo>=</mo><msub><mi>h</mi><mtext>m</mtext></msub><msub><mi>ρ</mi><mtext>s</mtext></msub><mi>ln</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>P</mi><mo>−</mo><msub><mi>P</mi><mrow><mtext>vi</mtext></mrow></msub></mrow><mrow><mi>P</mi><mo>−</mo><msub><mi>P</mi><mtext>v</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> The momentum-conservation equations are written with the following simplifications: (1) The Reynolds stresses are neglected. (2) The phase pressures are assumed to be equal and the interfacial pressures are equal to the phase pressures. (3) The interfacial momentum storage and phase viscous stresses are neglected. (4) The interface force terms consist of both pressure and viscous stresses. We thus write <math display="block" id="M7"><mtable><mtr><mtd><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><mi>A</mi><mfrac><mrow><mo>∂</mo><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><mi>A</mi><mfrac><mrow><mo>∂</mo><msubsup><mi>v</mi><mtext>g</mtext><mtext>2</mtext></msubsup></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><msub><mi>α</mi><mtext>g</mtext></msub><mi>A</mi><mfrac><mrow><mo>∂</mo><mi>P</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>+</mo><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><msub><mi>B</mi><mtext>x</mtext></msub><mi>A</mi><mo>−</mo><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><msub><mi>f</mi><mrow><mtext>wg</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>Γ</mi><mtext>g</mtext></msub><mi>A</mi><mrow><mo>(</mo><mrow><msub><mi>v</mi><mrow><mtext>gi</mtext></mrow></msub><mo>−</mo><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><msub><mi>f</mi><mrow><mtext>ig</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>v</mi><mtext>g</mtext></msub><mo>−</mo><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow><mo>−</mo><mi>C</mi><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>m</mtext></msub><mi>A</mi><mrow><mo>[</mo><mrow><mfrac><mrow><mo>∂</mo><mrow><mo>(</mo><mrow><msub><mi>v</mi><mtext>g</mtext></msub><mo>−</mo><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><msub><mi>v</mi><mtext>f</mtext></msub><mfrac><mrow><mo>∂</mo><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>−</mo><msub><mi>v</mi><mtext>g</mtext></msub><mfrac><mrow><mo>∂</mo><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac></mrow><mo>]</mo></mrow><mo>,</mo></mtd></mtr></mtable></math> <math display="block" id="M8"><mtable><mtr><mtd><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><mi>A</mi><mfrac><mrow><mo>∂</mo><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><mi>A</mi><mfrac><mrow><mo>∂</mo><msubsup><mi>v</mi><mtext>f</mtext><mn>2</mn></msubsup></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><msub><mi>α</mi><mtext>f</mtext></msub><mi>A</mi><mfrac><mrow><mo>∂</mo><mi>P</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>+</mo><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><msub><mi>B</mi><mi>x</mi></msub><mi>A</mi><mo>−</mo><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><msub><mi>f</mi><mrow><mtext>wf</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>Γ</mi><mtext>g</mtext></msub><mi>A</mi><mrow><mo>(</mo><mrow><msub><mi>v</mi><mrow><mtext>fI</mtext></mrow></msub><mo>−</mo><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><msub><mi>f</mi><mrow><mtext>if</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>v</mi><mtext>f</mtext></msub><mo>−</mo><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow><mo>−</mo><mi>C</mi><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>m</mtext></msub><mi>A</mi><mrow><mo>[</mo><mrow><mfrac><mrow><mo>∂</mo><mrow><mo>(</mo><mrow><msub><mi>v</mi><mtext>f</mtext></msub><mo>−</mo><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><msub><mi>v</mi><mtext>g</mtext></msub><mfrac><mrow><mo>∂</mo><msub><mi>v</mi><mtext>f</mtext></msub></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>−</mo><msub><mi>v</mi><mtext>f</mtext></msub><mfrac><mrow><mo>∂</mo><msub><mi>v</mi><mtext>g</mtext></msub></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> The momentum equations are unchanged when a noncondensable gas is present. f wg / f wf is the wall drag coefficient and f ig / f if is the interphase drag coefficient for the gas/liquid [ 14 14 ]. The gas-phase properties are evaluated for the steam/noncondensable-gas mixture in all of the equations. The energy-conservation equations are written as <math display="block" id="M9"><mtable><mtr><mtd><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi>g</mi></msub><msub><mi>ρ</mi><mi>g</mi></msub><msub><mi>U</mi><mi>g</mi></msub></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi>g</mi></msub><msub><mi>ρ</mi><mi>g</mi></msub><msub><mi>U</mi><mi>g</mi></msub><msub><mi>v</mi><mi>g</mi></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>P</mi><mfrac><mrow><mo>∂</mo><msub><mi>α</mi><mi>g</mi></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mi>P</mi><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi>g</mi></msub><msub><mi>v</mi><mi>g</mi></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mi>w</mi><mi>g</mi></mrow></msub><mo>+</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>g</mi></mrow></msub><mo>+</mo><msub><mi>Γ</mi><mrow><mi>i</mi><mi>g</mi></mrow></msub><msubsup><mi>h</mi><mi>g</mi><mo>*</mo></msubsup></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>Γ</mi><mi>w</mi></msub><msubsup><mi>h</mi><mi>g</mi><mo>'</mo></msubsup><mo>−</mo><msub><mi>Q</mi><mrow><mi>g</mi><mi>f</mi></mrow></msub><mo>+</mo><mi>D</mi><msub><mi>i</mi><mi>g</mi></msub><mo>,</mo></mtd></mtr></mtable></math> <math display="block" id="M10"><mtable><mtr><mtd><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><msub><mi>U</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><msub><mi>U</mi><mtext>f</mtext></msub><msub><mi>v</mi><mtext>f</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>P</mi><mfrac><mrow><mo>∂</mo><msub><mi>α</mi><mtext>f</mtext></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mi>P</mi><mi>A</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>v</mi><mtext>f</mtext></msub><mi>A</mi></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mtext>wf</mtext></mrow></msub><mo>+</mo><msub><mi>Q</mi><mrow><mtext>if</mtext></mrow></msub><mo>+</mo><msub><mi>Γ</mi><mrow><mtext>if</mtext></mrow></msub><msubsup><mi>h</mi><mtext>f</mtext><mtext>*</mtext></msubsup></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>Γ</mi><mtext>w</mtext></msub><msubsup><mi>h</mi><mtext>f</mtext><mo>'</mo></msubsup><mo>−</mo><msub><mi>Q</mi><mrow><mtext>gf</mtext></mrow></msub><mo>+</mo><mi>D</mi><msub><mi>i</mi><mtext>f</mtext></msub><mo>.</mo></mtd></mtr></mtable></math> The term Q gf in the above equations is defined as the sensible heat-transfer rate unit volume at the noncondensable-gas–liquid interface, and is given by <math display="block" id="M11"><mrow><msub><mi>Q</mi><mrow><mtext>gf</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mtext>n</mtext></msub></mrow><mi>P</mi></mfrac><msub><mi>H</mi><mrow><mtext>gf</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mtext>g</mtext></msub><mo>−</mo><msub><mi>T</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><mo>.</mo></mrow></math> The phase-wall heat-transfer rates per unit volume Q wg and Q wf are defined as <math display="block" id="M12"><mrow><msub><mi>Q</mi><mrow><mtext>wg</mtext></mrow></msub><mo>=</mo><msub><mi>H</mi><mtext>g</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>w</mtext></msub><mo>−</mo><msub><mi>T</mi><mrow><mtext>refg</mtext></mrow></msub></mrow><mo>)</mo></mrow><mtable><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable><msub><mi>Q</mi><mrow><mtext>wf</mtext></mrow></msub><mo>=</mo><msub><mi>H</mi><mtext>f</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>w</mtext></msub><mo>−</mo><msub><mi>T</mi><mrow><mtext>reff</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> The interfacial heat-transfer ( Q ig ) and mass-transfer (Г ig ) terms must be modified because of the presence of a noncondensable gas; that is, <math display="block" id="M13"><mrow><msub><mi>Q</mi><mrow><mtext>ig</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mtext>s</mtext></msub></mrow><mi>P</mi></mfrac><msub><mi>H</mi><mrow><mtext>ig</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>T</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><msub><mi>Γ</mi><mtext>w</mtext></msub><mrow><mo>(</mo><mrow><msubsup><mi>h</mi><mtext>g</mtext><mo>'</mo></msubsup><mo>−</mo><msubsup><mi>h</mi><mtext>f</mtext><mo>'</mo></msubsup></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> <math display="block" id="M14"><mrow><msub><mi>Γ</mi><mrow><mtext>ig</mtext></mrow></msub><mo>=</mo><mo>−</mo><msub><mi>Γ</mi><mrow><mtext>if</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mfrac><mrow><msub><mi>P</mi><mtext>s</mtext></msub></mrow><mi>P</mi></mfrac><msub><mi>H</mi><mrow><mtext>ig</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>T</mi><mtext>g</mtext></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>H</mi><mrow><mtext>if</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>T</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mi>h</mi><mtext>g</mtext><mo>*</mo></msubsup><mo>−</mo><msubsup><mi>h</mi><mtext>f</mtext><mtext>*</mtext></msubsup></mrow></mfrac><mo>,</mo></mrow></math> where h g * is the steam enthalpy based on the partial pressure of steam instead of the total pressure. The phase energy-dissipation terms Di g /Di f are decided by wall friction and pump effects. However, the dissipation effects due to interface mass transfer, interface friction, and virtual mass are neglected in the energy equations [ 15 15 - 17 17 15-17 ]. The dissipation effects are given as <math display="block" id="M15"><mrow><mi>D</mi><msub><mi>i</mi><mtext>g</mtext></msub><mo>=</mo><msub><mi>α</mi><mtext>g</mtext></msub><msub><mi>ρ</mi><mtext>g</mtext></msub><msub><mi>f</mi><mrow><mtext>wg</mtext></mrow></msub><msubsup><mi>v</mi><mtext>g</mtext><mtext>2</mtext></msubsup><mo>,</mo><mtext>  </mtext><mi>D</mi><msub><mi>i</mi><mtext>f</mtext></msub><mo>=</mo><msub><mi>α</mi><mtext>f</mtext></msub><msub><mi>ρ</mi><mtext>f</mtext></msub><msub><mi>f</mi><mrow><mtext>wf</mtext></mrow></msub><msubsup><mi>v</mi><mi mathvariant="normal">f</mi><mn>2</mn></msubsup><mo>.</mo></mrow></math> We next consider wall Condensation with a noncondensable gas, and note that condensation heat transfer at the wall plays a key role in determining the pressure transient in the steam-gas pressurizer. The following assumptions apply to our condensation model [ 18 18 ]: (1) The sensible heat transfer through the diffusion layer to the interface is negligible. (2) The gas is not removed from the vapor region by dissolving it in the condensate. (3) Stratification of the noncondensable gas in steam by buoyancy effects is negligible. The condensation heat and mass transfer are evaluated by Shah’s correlations [ 19 19 ] as follows: <math display="block" id="M16"><mrow><msub><mi>h</mi><mtext>c</mtext></msub><mo>=</mo><msub><mi>h</mi><mrow><mtext>sf</mtext></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>3.8</mn></mrow><mrow><msup><mi>Z</mi><mrow><mn>0.95</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo><mo>,</mo><mtext>  </mtext><mi>Z</mi><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>X</mi><mtext>m</mtext></msub></mrow></mfrac><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>0.8</mn></mrow></msup><msubsup><mi>P</mi><mrow><mtext>red</mtext></mrow><mrow><mn>0.8</mn></mrow></msubsup><mo>,</mo></mrow></math> <math display="block" id="M17"><mrow><msub><mi>h</mi><mrow><mtext>sf</mtext></mrow></msub><mo>=</mo><msub><mi>h</mi><mtext>l</mtext></msub><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>X</mi><mtext>m</mtext></msub></mrow><mo>)</mo></mrow></mrow><mrow><mn>0.8</mn></mrow></msup><mo>,</mo><mtext>  </mtext><msub><mi>h</mi><mtext>l</mtext></msub><mo>=</mo><mn>0.023</mn><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>k</mi><mtext>f</mtext></msub></mrow><mrow><msub><mi>D</mi><mtext>h</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><msubsup><mrow><mi>Re</mi></mrow><mtext>f</mtext><mrow><mn>0.8</mn></mrow></msubsup><msubsup><mrow><mi>Pr</mi></mrow><mtext>f</mtext><mrow><mn>0.4</mn></mrow></msubsup><mo>,</mo><mtable><mtr><mtd><mrow/></mtd></mtr></mtable><msub><mrow><mi>Re</mi></mrow><mtext>f</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>ρ</mi><mtext>f</mtext></msub><mi>v</mi><mi>D</mi></mrow><mrow><msub><mi>μ</mi><mtext>f</mtext></msub></mrow></mfrac><mi>，</mi><mtable><mtr><mtd><mrow/></mtd></mtr></mtable><msub><mrow><mi>Pr</mi></mrow><mtext>f</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mtext>f</mtext></msub><msub><mi>C</mi><mtext>p</mtext></msub></mrow><mi>k</mi></mfrac><mtext>,</mtext></mrow></math> where P red is the reduced bulk pressure; that is, <math id="M18"><mrow><msub><mi>P</mi><mrow><mtext>red</mtext></mrow></msub><mo>=</mo><mrow><mi>P</mi><mo>/</mo><mrow><msub><mi>P</mi><mrow><mi>c</mi><mi>r</mi></mrow></msub></mrow></mrow></mrow></math> . X is the gas-mixture (including steam and noncondensable gas) mass fraction in the pressurizer, where <math display="block" id="M19"><mrow><msub><mi>X</mi><mi>m</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>M</mi><mrow><mtext>mix</mtext></mrow></msub></mrow><mrow><msub><mi>M</mi><mrow><mtext>mix</mtext></mrow></msub><mo>+</mo><msub><mi>M</mi><mrow><mtext>liq</mtext></mrow></msub></mrow></mfrac><mo>.</mo></mrow></math> The effects of noncondensable gas are represented by multipliers that modify and reduce the heat-transfer coefficient. The modification factor is f c , which is derived from the Vierow-Schrock correlation [ 20 20 ]: <math display="block" id="M20"><mrow><msub><mi>f</mi><mtext>c</mtext></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mn>1</mn><mo>−</mo><mn>10</mn><msub><mi>X</mi><mtext>n</mtext></msub><mo>,</mo><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><msub><mi>X</mi><mtext>n</mtext></msub><mo><</mo><mn>0.063</mn></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mo>−</mo><mn>0.938</mn><msubsup><mi>X</mi><mtext>n</mtext><mrow><mtext>0</mtext><mtext>.13</mtext></mrow></msubsup><mo>,</mo><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><mn>0.063</mn><mo>≤</mo><msub><mi>X</mi><mtext>n</mtext></msub><mo>≤</mo><mn>0.60</mn></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mo>−</mo><msubsup><mi>X</mi><mtext>n</mtext><mrow><mtext>0</mtext><mtext>.13</mtext></mrow></msubsup><mo>,</mo><mtable><mtr><mtd><mrow/></mtd><mtd><mrow><msub><mi>X</mi><mtext>n</mtext></msub><mo>></mo><mn>0.60</mn></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></mrow><mo>.</mo></mrow></math> The saturation temperature at the interface between the steam and water film is solved by the Colburn-Hougen diffusion calculation [ 21 21 ]. 3 RESULTS AND DISCUSSION 1 3.1 Experimental Apparatus 2 The results from MIT insurge experiments were used to verify the steam-gas pressurizer model in this study. The experimental apparatus is shown in Fig. 2. Fig. 2. The test facility essentially consisted of two stainless-steel tanks, an insurge line, and a gas-injection system. The primary tank, which models the pressurizer volume, is a 203.2-mm-inner-diameter pipe and is 1143 mm in height. The storage tank is filled with ~20-°C water pressurized with nitrogen. The noncondensable-gas-injection line is in the bottom of the primary tank. In conjunction with the ~20-°C insurge water, the operating pressure in the primary tank was brought up to approximately 0.53 MPa. Steam was bubbled through the primary tank to assist the heaters in bringing the tank up to saturation conditions. After the tank reached saturation, the steam line was isolated, and then the noncondensable gas was injected into the vapor space after the primary tank had been purged of other dissolved gases, and before the system had reached operating pressure. In order to reduce local density gradients in the primary tank during the injection of a noncondensable gas, the process of injection was carried out as slowly as possible. The system was then brought up to about 0.53 MPa with the noncondensable gas already introduced into the primary tank. When the tank reached operating pressure and the system was at thermal equilibrium, the saturation conditions in the tank were maintained by immersion heaters; the heaters were turned off when the insurge was initiated. Before the insurge experiment began, the initial water level in the primary tank was kept at 413.8 mm. The insurge was then initiated by opening the quick-opening valve at the base of the pressurizer; the insurge rate was approximately 0.4 kg/s. Based on the MIT experimental apparatus and conditions, the nodes of the experimental system are shown in Fig. 3. Fig. 3. The following simplifications were made in Relap5 model: The storage tank was represented by a time-dependent volume component, designated the 110 card, and the valve and pipe between the primary tank and storage tank were represented by a time-dependent junction component, designated the 120 card, to regulate the insurge flow rate. The primary tank was modeled as a pipe component, designated the 108 card. The primary tank and heat structure were divided into 10 control volumes, which was the best choice based on a sensitivity analysis. During the simulation, the initial condition in the 110 card was set to 20 °C and the insurge rate, 0.4kg/s, was set in the 120 card. The primary tank, the 108 card, was filled with water to a level of 413.8 mm and vapor with noncondensable gas. The mass fraction of and type of noncondensable gas was changed during the simulation, and the heat structure was used to simulate the environmental heat loss, 1.3 kW. The calculations were performed using Relap5, and nitrogen and argon were chosen as the noncondensable gases for modeling. 3.2 Results 2 Figure 4 Figure 4 shows the calculation results without a noncondensable gas in the pressurizer. The initial pressure in the primary tank was 0.531 MPa. Upon the insurge flow entering the primary tank the pressure in the pressurizer changed significantly. The peak pressure was 0.647 MPa when the insurge ended. Subsequently, the pressure in the pressurizer decreased because of the energy dissipation. Inspection of Fig. 4 Fig. 4 indicates that the calculations of the model in this study are in excellent agreement with the experimental results when there is no noncondensable gas in the pressurizer; the maximum error in the calculation is 0.8%. Thus, the model can accurately predict the pressure in the pressurizer without a noncondensable gas. The distribution of temperature in the pressurizer is shown in Fig. 5. Fig. 5. Since the insurge water temperature was 20 °C, which was significantly cooler than the initial temperature in the pressurizer, thermal stratification occurs as is shown in the figure. The liquid region (at 0.0254, 0.1676, and 0.5740 m) in the pressurizer experienced an obvious change in temperature. The steam/gas region (at 1.00 m) exhibited a very small temperature variation. The temperature change became smaller with increasing height, and the largest temperature drop (approximately 125 °C) occurred in the bottom of the primary tank. It is easily found from inspection of Fig. 5 Fig. 5 that there was an abrupt temperature drop at 1.00 m. Owing to the insurge of water, the pressure in the steam-gas pressurizer will increase, and then the temperature will also increase with the change of pressure because of the saturation conditions in the steam/gas region. An abrupt drop in pressure will occur when the insurge ends (after approximately 33 s, as shown in Fig. 5 Fig. 5 ), so the saturation temperature will decrease abruptly as well, also shown in Fig. 5. Fig. 5. The calculation results for different mass fractions of nitrogen are shown in Fig.6. Fig.6. The change of pressure in the pressurizer is apparently different when the mass fraction of nitrogen varies, and the peak pressure increases with the increase of nitrogen mass fraction. The peak pressure ranged from 0.71 to 1.045 MPa when the mass fraction of nitrogen ranged from 3% to 20%. This is due to the condensation heat transfer in the presence of a non-condensable gas, which decreases the heat-transfer coefficient with increasing mass fraction of a noncondensable gas. We note that the Relap5 model calculations have same development tendency as the experimental results when the nitrogen mass fraction varies, while the calculation errors are different. For 3% N 2 , the Relap5 model performs much better, and the predicted and experimental values are in excellent agreement with each other and the errors are rather small. For 10% and 20% N 2 , the calculation accuracy after the peak pressure is reached becomes worse but still tolerable, with the maximum error being approximately 13.1% at a time of 45 s for 10% N 2 . However, the model can accurately predict the pressure change before the insurge ends, and the peak pressures are therefore able to be calculated accurately. The pressure transients in the pressurizer when the noncondensable gas is argon are shown in Fig.7 Fig.7 , which indicates that the model gives good calculation accuracy for small mass fractions of argon. As expected, as the argon mass fraction increases, the errors of the Relap5 model also increase. The peak pressure values are well predicted, but after reaching peak pressure the pressure values predicted by the Relap5 model tend to be smaller compared to experimental values; the maximum error is 10%. It can be realized from Figs. 6 Figs. 6 and 7 7 that the pressure transients can be predicted accurately by the Relap5 model developed for this study, especially for the pressurizer’s peak pressure. The pressure transients in the pressurizer are very different for different types of noncondensable gas. As shown in Fig. 8 Fig. 8 , there is an obvious difference between nitrogen and argon with the same mass fraction (10%). The pressure in the presence of nitrogen is always larger than that in the presence of argon, and the peak pressure difference reached 0.068 MPa in our experiment. The difference in peak pressure is caused by the degradation of condensation heat transfer due to the different types of non-condensable gas, either nitrogen or argon. It is easily seen that the pressure for nitrogen will always be larger than that of argon at other mass fractions. Comparing the Relap5 model calculations with the experimental values, it can be seen that the noncondensable gas plays a key role in the pressure transient process in the steam-gas pressurizer, and it is therefore necessary to consider the effect of the noncondensable gas in the steam-gas pressurizer model. Indeed, even a small amount of noncondensable gas can considerably reduce the condensation heat transfer near the wall. In this study, a modification factor f c for the wall condensation model was introduced to evaluate the effect of a noncondensable gas on the heat- and mass-transfer processes during vapor condensation near the wall. However, this method sometimes seems a little rough and the model’s calculation accuracy may be unsatisfactory but still tolerable when the mass fraction of the noncondensable gas is large. 4 CONLUSION 1 In this study, a steam-gas pressurizer model was developed based on Relap5 code and verified using the results of MIT insurge experiments. The model well predicted the pressure transient in the steam-gas pressurizer, and the effect of a noncondensable gas on the condensation near the wall was considered under different experimental conditions. When the mass fraction of the noncondensable increased, the peak pressure increased considerably. At the same time, the pressure change process may be different due to different types of noncondensable gas. References Y.J. Zhang , X.Z. Wang , Development progress and pros-pect for industrial application of the 200MW low temperature nuclear heating reactor . Nuclear Power Engineering , 2003 , 24 ( 2 ): 180 - 183 . (in Chinese) Development progress and pros-pect for industrial application of the 200MW low temperature nuclear heating reactor O.B. Samoilov , A.V. Kurachenkov , Nuclear district heating plants AST-500 . Nucl Eng Des , 1997 , 173 ( 1 ): 109 - 117 . DOI: 10.1016/S0029-5493(97)00087-3 http://doi.org/10.1016/S0029-5493(97)00087-3 Nuclear district heating plants AST-500 T.-W. Kim , J.-W. Kim , G.-C Park . Development of nonequ-ilibrium pressurizer model with noncondensable gas . Nucl Eng Des , 2006 , 236 ( 4 ): 375 - 384 . DOI: 10.1016/j.nucengdes.2005.09.003 http://doi.org/10.1016/j.nucengdes.2005.09.003 Development of nonequ-ilibrium pressurizer model with noncondensable gas M.T. Leonard , P. Griffith , The effects of a noncondensable gas on pressurizer insurge transients . http://www.researchgate.net/publication/37600723_The_effects_of_a_noncondensable_gas_on_pressurizer_insurge_transients http://www.researchgate.net/publication/37600723_The_effects_of_a_noncondensable_gas_on_pressurizer_insurge_transients The effects of a noncondensable gas on pressurizer insurge transients S.N. Kim , P. Griffith , PWR pressurizer modeling . Nucl Eng Des . 1987 , 102 ( 2 ): 199 - 209 . DOI: 10.1016/0029-5493(87)90253-6 http://doi.org/10.1016/0029-5493(87)90253-6 PWR pressurizer modeling M. Murase , Y. Kataoka , T. Fujii , Evaporation and condensa-tion heat transfer with a noncondensable gas present . Nucl Eng Des , 1993 , 141 ( 1 ): 135 - 143 . DOI: 10.1016/00295493(93)9098-T http://doi.org/10.1016/00295493(93)9098-T Evaporation and condensa-tion heat transfer with a noncondensable gas present T.W. Kim , Development of steam-gas pressurizer model based on two-region nonequilibrium with noncondensable gas. Ph. D. Thesis , Seoul National University , Korea , 2005 . Development of steam-gas pressurizer model based on two-region nonequilibrium with noncondensable gas. Ph. D. Thesis G. Espinosa-Paredes , A.N. Ez-Carrera , SBWR Model for Steady-State and Transient Analysis . Science and Technology of Nuclear Installations , 2008 , 01 : 1 - 18 . DOI: 10.1155/2008/428168 http://doi.org/10.1155/2008/428168 SBWR Model for Steady-State and Transient Analysis G. Espinosa-Paredes , A. Nuñez-Carrera , A. Vazquez-Rodriguez , Simplified distributed parameters BWR dynamic model for transient and stability analysis . Annals of Nuclear Energy , 2006 , 33 ( 33 ): 1245 - 1259 . DOI: 10.1016/j.anucene.2006.08.009 http://doi.org/10.1016/j.anucene.2006.08.009 Simplified distributed parameters BWR dynamic model for transient and stability analysis G. Espinosa-Paredes , J. Alvarez-Ramirez , A. Nunez-Carrera , et al . Dynamic Comparison of Three and Four-Equation Reactor Core Models in a Full-Scope Power Plant Training Simulator . Nuclear Technology , 2004 , 145 ( 2 ): 150 - 162 . http://www.ans.org/pubs/journals/nt/a_3466 http://www.ans.org/pubs/journals/nt/a_3466 Dynamic Comparison of Three and Four-Equation Reactor Core Models in a Full-Scope Power Plant Training Simulator S.M. Baek , H.C. No , I.Y. Park , A nonequilibrium three-region model for transient analysis of pressurized water reactor pressurizer . Nucl Technol , 1986 , 74 ( 3 ): 260 - 266 . http://www.ans.org/pubs/jo-urnals/nt/a_33828 http://www.ans.org/pubs/jo-urnals/nt/a_33828 A nonequilibrium three-region model for transient analysis of pressurized water reactor pressurizer C.Q. Yan , Two Phases Flow . Harbin (China) : Harbin Enginee-ring University Press , 2010 , 30 - 38 . (in Chinese) Two Phases Flow S. Patankar , Numerical heat transfer and fluid flow . USA, CRC Press , 1980 , 41 - 109 Numerical heat transfer and fluid flow J.H. Lienhard , A heat transfer textbook . USA, CourierCorporation , 2013 , 399 - 416 A heat transfer textbook G. Espinosa-Paredes , O. Cazarez-Candia , A. Vazquez , Theoretical derivation of the interaction effects with expansion effects to bubbly two-phase flows . Annals of Nuclear Energy , 2004 , 31 ( 2 ): 117 - 133 . DOI: 10.1016/S0306-4549(03)00218-4 http://doi.org/10.1016/S0306-4549(03)00218-4 Theoretical derivation of the interaction effects with expansion effects to bubbly two-phase flows G. Espinosa-Paredes , A. Nuñez-Carrera , E.J. Martinez-Mendez , Interaction forces model on a bubble growing for nuclear best estimate computer codes . Annals of Nuclear Energy , 2005 , 32 ( 14 ): 1546 - 1566 . DOI: 10.1016/j.anucene.2005.04.002 http://doi.org/10.1016/j.anucene.2005.04.002 Interaction forces model on a bubble growing for nuclear best estimate computer codes F.J. Valdes-Parada , G. Espinosa-Paredes , Darcy's law for immiscible two-phase flow: A theoretical development . Journal of Porous Media , 2005 , 8 : 557 - 567 . DOI: 10.1615/JPorMedia.v8.i6.20 http://doi.org/10.1615/JPorMedia.v8.i6.20 Darcy's law for immiscible two-phase flow: A theoretical development M.M. Shah , A general correlation for heat transfer during film con-densation inside pipes . INT J HEAT MASS TRAN , 1979 , 22 ( 4 ): 547 - 556 . DOI: 10.1016/0017-9310(79)90058-9 http://doi.org/10.1016/0017-9310(79)90058-9 A general correlation for heat transfer during film con-densation inside pipes K.Y. Lee , M.H. Kim , Experimental and empirical study of steam condensation heat transfer with a noncondensable gas in a small-diameter vertical tube . Nucl Eng Des , 2008 , 238 ( 1 ): 207 - 216 . DOI: 10.1016/j.nucengdes.2007.07.001 http://doi.org/10.1016/j.nucengdes.2007.07.001 Experimental and empirical study of steam condensation heat transfer with a noncondensable gas in a small-diameter vertical tube K.M. Vierow , V.E. Schrock , Condensation in a natural circulation loop with noncondensable gases, 1 , Proceedings of the intern ati-onal conference on multiphase flows' 91 , Tsukuba, Japan , 1991 . Condensation in a natural circulation loop with noncondensable gases, 1 A.P. Colburn , O.A. Hougen , Design of cooler condensers for mix-tures of vapors with noncondensing gases . Ind Eng Chem , 1934 , 26 ( 11 ): 1178 - 1182 . DOI: 10.1021/ie50299a011 http://doi.org/10.1021/ie50299a011 Design of cooler condensers for mix-tures of vapors with noncondensing gases K.E. Carlson , R.A. Riemke , S.Z. Rouhani S Z , et al . RELAP5/MOD3 Code Manual Volume I: Code Structure, System Models and Solution Methods. US NRC NUREG/CR-5535 , 1990 . RELAP5/MOD3 Code Manual Volume I: Code Structure, System Models and Solution Methods. US NRC NUREG/CR-5535 10.1007/s41365-017-0223-x 2015-08-21 2015-12-11 2015-12-18 2017-03-28 2017-05-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2017 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-017-0223-x http://dx.doi.org/10.1007/s41365-017-0223-x . 2017 NOMENCLATURE A area, m 2 B body force m/s 2 C coefficient of virtual mass D diameter, m Di energy dissipation function, W/m 3 F drag coefficient, m 3 /kg·s f c modification factor H volumetric heat transfer coefficient, W/K · m 3 h specific enthalpy, J/kg h c condensation heat transfer coefficient with non-condensable gas, W/(m 2 ·K) h l Dittus-Boelter coefficient assuming all fluid is liquid, W/(m 2 ·K) h sf superficial heat transfer coefficient, W/(m 2 ·K) k thermal conductivity, W/m·K M mass, kg P pressure, pa P r Prandtl number P red reduced bulk pressure, pa Q volumetric heat rate, W/m 3 Re Reynolds number T temperature, K t time, s U specific internal energy, J/kg v velocity, m/s X mass fraction Z two phase friction correlation factor Symbols α void fraction Г volumetric mass exchange rate, kg/m 3 ·s ε coefficient ρ density, kg/m 3 Subscripts cr critical condition f liquid phase g gas phase h hydraulic i interface liq liquid phase m mixture of gas phase and liquid phase mix mixture of vapor and non-condensable gas n non-condensable gas ref reference condition s steam component of gas phase w wall Superscripts * Total derivative of saturation property with respect to pressure ' derivative Nuclear Science and Techniques 05 28