1 INTRODUCTION

Cryogenic flow systems that include two-phase flow are widely used in liquefied natural gas plants, aerospace technology, superconductivity technology, and many other engineering applications. The investigation of the boiling two-phase flow characteristics of cryogenic fluids is very interesting and important not only in the basic study of the hydrodynamics of cryogenic fluid, but also in providing solutions to problems related to practical engineering applications of cryogenic two-phase flow [1–3]. Cryogenic fluids are characterized by large compressibility compared with fluids at room temperature such as water, a small difference in density between gas and liquid phases, and a small latent heat of vaporization. These unique characteristics of cryogenic contribute to obtain high performance in fluid apparatuses, such as two-phase operation of inducers. Pool boiling heat transfer is a subject of extensive research worldwide. A large number of both theoretical and empirical correlations have been proposed to predict the pool boiling heat transfer curve. The nucleate pool boiling curve is dependent on many parameters, mainly related to the thermo-physical properties of the boiling fluid and to the characteristics of the surface on which the pool boiling takes place. In particular the effect of the surface characteristics such as the thermo-physical properties of the material, geometry, thickness, orientation, surface conditions (roughness, oxidation and even the boiling cycles), require further investigation [4]. The mechanism of pool boiling can be extraordinarily complicated and far from fully developed, especially for the cryogenic liquids. Bubble dynamics in pool boiling, which has been intensively studied since Rayleigh first formulated a correlation for vapor bubble, plays a significant role both in experimental analyses and the corresponding numerical simulations. The following bubble nucleation quantities are usually considered in matching model predictions to experimental observations [5, 6]: a) average bubble departure diameter—\(d_{bw};\) b) average bubble departure frequency—\(f_{d};\) c) average active nucleation site density—\(N_{A}\). Although a considerable amount of literature dealt with pool boiling phenomena with water, organic fluids, and refrigerants through experiments or CFD models, counterpart researches using cryogenic liquids are far from enough both theoretically and experimentally. Due to the small latent heat of vaporization and surface tension, nucleation sites and departure of nitrogen bubble have different characteristics [2]. To further understand the nucleate pool boiling of \(LN_{2}\), new correlations in some works [7] are established and improved in accordance to the experimental results. This correlations are used in the forthcoming numerical simulations, which are verified by evaluating temperatures, distributed on heated wall and compared with experimental data. There are a number of studies reported in the literature on the numerical modeling of subcooled boiling. According to the widely used heat flux partitioning model [8, 9], the boiling heat transfer is divided into three components; convective, quenching and evaporative heat fluxes. The convective part is for the single-phase convection, quenching refers to liquid filling the wall vicinity after bubble detachment due to vaporization and the evaporative component is for the fluid that evaporates. The model developed in [7, 9], as wall heat flux partitioning model has been used widely along with various correlations that are available in the literature for bubble departure diameter, bubble departure frequency and nucleation site density. In paper [10] simulated boiling low of nitrogen using a two-dimension drift-flux model which is considered to be a simplified two-fluid model. An Eulerian–Eulerian two-phase flow model has been developed to simulate the boiling heat transfer phenomenon in a pipe flow [6]. In [12] established closure correlations in describing bubble nucleation and departure on the heating surface and a modified two-fluid model was introduced into simulating boiling low of liquid nitrogen in a vertical tube. While most efforts were focused on development of commercial or in-house codes, the present work is aimed at developing further the open source numerical code to simulate boiling flows under different operating conditions and verify the predictions using the data available in the literature [11]. The other modifications of code was used in various applications very successfully to solve different multiphase flows [13–16]. The model was implemented in the OpenFOAM based source code [17] for modelling boiling of liquid water in different systems. The code development process was divided into two sections. In the first step, an adiabatic two-phase flow model which takes into account the effect of interfacial forces was developed. In the second step, the energy equation was activated to account for non-adiabatic conditions. In order to include the boiling effect, several different subroutines which model evaporation and condensation phenomena were attached to the solver. Results of the two-phase adiabatic flow and from the boiling model are compared with available numerical and experimental data. The simulation predictions are in reasonable agreement with the experimental data and show significant improvement relative to previous numerical results, which suggests the validity of the developed model for boiling heat transfer problems.

2 MODEL DESCRIPTION

In a two-fluid Eulerian–Eulerian model, each phase is described by a set of averaged conservation equations, and the continuity, momentum and energy equations are solved for each phase. These, being discussed for two-phase boiling flows in previous publication [14, 17], to which the interested reader may refer to, are not presented here. As a consequence of the averaging procedure, details of the interphase structure are lost and closure models are required for the mass, momentum and energy transfers at the interphase. When boiling occurs at a heated wall, different heat transfer mechanisms take place and these need to be modeled. In regions of the wall where no bubbles are growing, the heat is transferred to the liquid by single-phase convection. Otherwise, the heat is removed by the evaporation process and supports the growth of bubbles at the nucleation sites. Bubbles grow attached to the wall until, when certain conditions are reached, detachment occurs. Detachment of bubbles promotes additional mixing in the fluid phase and the recirculation of subcooled liquid which is brought into contact with the wall to fill the volume which was previously occupied by the detaching bubble. This mechanism accounts for a portion of the heat transferred from the wall, and is known as quenching. Finally, when a significant amount of vapour is present at the wall, liquid access to the wall may be restricted and a portion of the heat is transferred by convection to the vapour phase. Therefore, the total heat transferred from the wall is partitioned between these heat transfer mechanisms [9, 18]:

$$Q_{w}=Q_{q}+Q_{e}+Q_{c},$$
(1)

where

$$Q_{q}=\sqrt{k_{l}\rho_{l}c_{pl}}\sqrt{f_{d}}A_{q}(T_{w}-T_{l}),\quad Q_{e}=\frac{\pi}{6}d_{bw}^{3}\rho_{l}f_{d}N_{a}H_{lg},$$
(2)
$$Q_{c}=St\rho_{l}c_{pl}\vec{u}_{l}(1-A_{q})(T_{w}-T_{l}),\quad A_{q}=N_{a}K\frac{\pi d_{bw}^{2}}{4},$$
(3)

\(k_{l}\), \(\rho_{l}\) and \(c_{pl}\) are coefficients of thermal conductivity, density and specific heat at constant pressure for the liquid phase. The \(\rho_{g}\) and \(H_{lg}\) are the bubble density and the latent heat of vaporization, \(f_{d}\) and \(A_{q}\) are the bubble departure frequency and the fraction of wall area subjected to cooling by quenching. The local tangential liquid velocity adjacent to the heated surface is \(u_{l}\). The wall temperature and fluid temperature near the wall are \(T_{w}\) and \(T_{l}\) correspondingly. The \(f_{d}\) and \(d_{bw}\) are the density of active nucleation sites and the bubble departure diameter. In this equation (2) the evaporative heat flux \(Q_{e}\) is known from the number of bubbles that grow attached to the heated wall at the active nucleation sites. These bubbles grow until the forces that promote detachment overcome those that keep the bubble attached to the wall. Therefore, the evaporative heat flux is known from the number of active nucleation sites \(N_{a}\), the diameter of the bubbles at departure \(d_{bw}\) and the frequency of the bubble departure from the wall \(f_{d}\). The parameter \(K\) is the ratio of the area around a nucleation site influenced by heat transported by a nucleate boiling to the projected bubble area at departure, whose value of 1.8 is taken from [7].

The model for \(N_{a}\) are used from [19] and is given by

$$N_{a}=\frac{\psi(\varrho^{*})R^{*-0.44}_{c}}{d_{bw}^{2}},$$
(4)

where

$$\varrho^{*}=\frac{\rho_{l}-\rho_{g}}{\rho_{g}},\quad R^{*}_{c}=\frac{4\sigma T_{sat}}{\Delta T_{sup}\rho_{g}H_{lg}d_{bw}},$$
(5)
$$\psi(\varrho^{*})=2.157\times 10^{-7}\varrho^{*-3.2}(1+0.0049\varrho^{*})^{4.13}.$$
(6)

The bubble departure frequency is calculated as

$$f_{d}=\sqrt{\frac{4}{3}\frac{g(\varrho_{l}-\varrho_{g})}{d_{bw}\varrho_{l}}}.$$
(7)

Object to the forces acting on the bubbles both axial and normal directions, at certain bubble size, gas vapor bubbles detach from the heated surface and flow along with the bulk fluid. In paper [18] analyzed the process of bubble departure according to fluid statics suggesting buoyancy and surface tension are in equilibrium when bubble detaches the heated surface. The relationship of bubble departure diameter for given conditions is thus expressed as

$$d_{bw}=\chi(\theta)\left[\frac{\sigma}{g(\rho_{l}-\rho_{g})}\right]^{0.5},\quad\chi(\theta)=0.208\theta,$$
(8)

where \(\sigma\) and \(\theta\) are surface tension and contact angle. Contact angle \(\theta\) is the key of the model correction. It influences the bubble shape within small pits and cavities on the heated surface, which is important to the correct simulation of bubble detachment from the surface. Then for liquid nitrogen, the bubble departure diameter is about 0.2 mm, which is smaller by one order of magnitude compared with that of water.

The overall model, implemented in the modified OpenFOAM code, is solved in a sectional three-dimensional geometry with symmetry conditions on both planes. Input and output boundary conditions are set at the input part and the output circular sector, the conditions of adhesion on the channel wall for the liquid and gas phase, heat flux on the part of the outer wall boundary, symmetry conditions for both phases on the side walls of the sector. At the inlet, fully-developed single-phase liquid velocity, turbulence and temperature are imposed, together with an imposed pressure at the outlet and the no-slip condition, and an imposed heat flux, at the wall. Strict convergence of residuals was ensured, together with a mass balance error always lower than 0.01\(\%\) for both phases.

3 RESULTS AND DISCUSSION

The verification of the presented mathematical model was carried out by comparing the results of calculations with experimental data. In the benchmark experiments [12], operating parameters including the outlet pressure, the heat flux and the mass flow rate are carefully selected. The outlet pressure \(P_{out}\) varies from \(0.195\) to \(1.129\) MPa and the mass flux \(G_{l}\) is in the range of \(184.7{-}584.8\) kg/(m\({}^{2}\) s). The inlet liquid subcooling \(\Delta T\) during the experiments appears in the range of \(0.87{-}10.61\) K. The heat flux \(q_{W}\) is in the range of \(1026{-}30125\) W m\({}^{-2}\). The data of experiments presented in [12] were chosen, in which the boiling ascending bubble flow of subcooled liquid at pressure in a tube with wall heating (diameter of the pipe \(D=6.0\) mm, height \(L=1000\) mm, height of the heated section 1000 mm) was studied. In the table the basic characteristics of gas-liquid flow at the inlet of the pipe are presented: \(G_{l}\) is the mass flow rate of liquid at the inlet of the pipe, \(P_{out}\) is the outlet pressure, \(q_{W}\) is the heat flux on the wall of the pipe, \(T_{2,IN}\) is the temperature of liquid at the inlet, \(\Delta T\) is the degree of subcooling of liquid. Two-phase flow in a vertical pipe is assumed to be axisymmetric, therefore for numerical simulation the calculated the area consisting of a circular sector with a radius of \(r_{0}=D/2=3.0\times 10^{-3}\) m, a length of \(L=1.0\) m and a solution angle of 5 degrees. Numerical calculations were performed on finite volume grids consisting of \(N_{e}=8000\), 16000, 32000 nodes of the computational grid. In the section of the plane \(x_{1}x_{2}\), the number of partitions by coordinates along the pipe axis and by length for different grids is \(N_{1}=10\times 100\), \(N_{2}=20\times 200\), \(N_{3}=40\times 400\), respectively. The numerical solution on the \(N_{3}\) grid differs from \(N_{2}\) less than by \(2\%\), so all the above numerical results are performed using a grid of finite volumes \(N_{2}\). The following parameters of the nitrogen-steam two-phase components are set: density \(\rho_{l}=806\) kg/m\({}^{3}\), \(\rho_{g}=4.6\) kg/m\({}^{3}\), dynamic nitrogen viscosity coefficient \(\mu_{l}=9.9\times 10^{-5}\) Pa s, vapour \(\mu_{g}=5.4\times 10^{-5}\) Pa s, the latent heat of vaporization is \(L_{lg}=2.2\times 10^{6}\) J/kg, constant pressure heat capacity \(C_{pl}=2042\) J/kg K, \(C_{pg}=1072\) J/kg K, coefficients of heat conductivity of environments \(k_{l}=0.72\) W/m K, \(k_{g}=0.15\) W/m K.

Table 1 Characteristics of mode parameters of experiments [12]
Full size table

The subcooled boiling is divided into three regions according to the fluid flow and heat transfer; including single phase region, highly subcooled region and slightly subcooled region.

The effects of the inlet mass flow rate, wall heat flux and inlet subcooled temperature on the distribution of wall temperature \(T_{w}\) along the axial direction are analyzed in this section. The distribution of \(T_{w}\) (numerical and experimental data) of liquid nitrogen along the wall of tube is shown in Fig. 1 for case 6 of experimental data [12] and on Fig. 2 for case 8 correspondingly. This both pictures shows the predicted temperature of wall versus different subcooled boiling flow conditions. The first picture (case 6) are at the stage of highly subcooled boiling, while the other (case 8) are at the stage of slightly subcooled boiling. The distributions of \(T_{w}\) along the axial direction are all nonlinear and the values all increase with length. For different parameters, the variation trends are different. The wall temperature increases with the wall more higher heat flux (Fig. 1) very rapidly and then slow down near the output of the tube. Otherwise, for lower heat flux (Fig. 2) distributions of \(T_{w}\) increases continuously until the output of the tube. The increase of the subcooled temperature causes an increase of the rate of bubble condensation in the bulk fluid and more bubbles vanish sequentially. The increase of inlet mass flow rate intensifies the convection heat transfer. The variation of the \(T_{w}\) with length exhibits a non-monotonic behavior (Fig. 1). At the initial region of subcooled boiling, the \(T_{w}\) increases gradually and then when evaporation is started the temperature is only very slightly becomes larger. For second case the \(T_{w}\) increase linearly with the tube length, that says that the evaporation process becomes not dominant here.

Fig. 1
figure 1

Comparison of predicted wall temperature \(T_{w}\) against experimental data for case 6 (1—line calculation,2—symbols \(\times\) experiment [12]).

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Fig. 2
figure 2

Comparison of predicted wall temperature \(T_{w}\) against experimental data for case 8 (1—line calculation,2—symbols \(\times\) experiment [12]).

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4 CONCLUSION

The two-fluid model of subcooled boiling low of liquid nitrogen was formulated by incorporating some correlations related to heat and mass transfer on the heating surface. Experimental cases in the literature were numerically simulated using the modified two-fluid model approach in the present study. When simulating subcooled boiling flow using a two-fluid model, one has to supply the closure correlations describing heat transfer from the heating surface to the fluids. Although lots of empirical or semi-empirical correlations have been established for subcooled boiling flow of high-boiling liquids, these correlations may be invalid, when modeling subcooled boiling flow of nitrogen, mainly due to the difference in physical properties of the liquids. Based on the current subcooled boiling phase model, the important bubble model parameters with respect to the boiling process were amended according to the mechanism of boiling heat transfer and unique physical properties of cryogenic fluids. In this job, some closure correlations for the bubble departure diameter, active site density and bubble waiting time are tested and numerical simulations are performed in the frame of the modified two-phase solver from OpenFOAM code. Benchmark experiments show that the correlations considering this difference achieve satisfactory agreement with the experimental data.