Keywords

1 Introduction

The passive heat removal system is one of the important systems to ensure the safety of ocean nuclear power plants. While using seawater as the infinite heat sink to cool the reactor, it also suffers the risk of reverse flow of subcooled seawater into the steam tube, which gives rise to the direct contact between the steam and subcooled water, and triggers condensation-induced water hammer (CIWH), resulting in high peak pressure pulses and consequently pipeline damage. CIWH is a common phenomenon in nuclear power plants, steam power plants, and other energy power plants, and has received widespread attention from domestic and international research institutions and scholars.

Regarding CFD numerical simulation methods for CIWH, there are three main types of representative models: one is the use of surface renewal theory to calculate condensation heat transfer, such as Tiselj [1], Strubelj [2], and Ceuca [3], which established a large interface model for direct contact condensation based on surface renewal theory, which considered the turbulence effect between the vapor-liquid phase and showed that the model can predict the transient temperature change better by comparing with the PMK-2 experimental data. Second, the temperature gradient of the vapor-liquid phase is calculated directly, and the vapor condensation is calculated based on the thermal conductivity heat transfer [4], and this method can also predict the fluid temperature variation of the direct contact condensation process better. Third, a traditional evaporative condensation model is used for the calculation, such as Pham et al. [5], who obtained the CIWH pressure wave at the MPa level through Lee model simulations. In addition, the two-fluid model used by Höhne et al. [6] and the Large Eddy Simulation (LES) method used by Li et al. [7] also allow the simulation of direct contact condensation processes with a good agreement with experiments. These method give visual access to two-phase flow and phase interface information, however, most of the results obtained by commercial CFD software are not very satisfactory in the calculation of CIWH pressure waves, and it is difficult to capture the peak pressure.

For the prediction of condensation-induced water hammer pressure waves, several research institutions and scholars have developed computational models specifically for CIWH, and representative ones include the WAHA3 numerical computational program developed by Barna [8] and Tiselj [9] based on the WAHALoads project, as well as the computational program ATHLET [10]. The above models use a one-dimensional, two-fluid, six-equation model, and a surge capture algorithm for pressure wave prediction of CIWH. Hibiki [11] also computationally captured the transient process of CIWH based on the RELAP5 code, but the validity of this approach is controversial. Milivojevic et al. [12] developed a one-dimensional, three-equation single fluid model based on the HEM (Homogeneous Equilibrium Model) model that captured the steam-filled CIWH phenomenon in a vertical pipe. However, this method has not been applied in CIWH calculations for stratified flow in horizontal pipes because it cannot characterize the heat mass transfer relationship between the gas and liquid phases.

2 Mathematical Model Development

2.1 Basic Control Equations

For the vapor-liquid two-phase flow model, the basic idea for establishing the control equation is that regard the two-phase mixed flow as two single-phase flow, and their characteristics in terms of continuity, momentum and energy are examined separately, and then the inter-phase effects of mass exchange, momentum exchange and energy exchange are taken into account to form the basic control equation of the two-phase flow.

The mass, momentum and energy balance equations for both phases are the following:

$$ \frac{{\partial A\alpha_{l} \rho_{l} }}{\partial t} + \frac{{\partial A\alpha_{l} \rho_{l} v_{l} }}{\partial x} = A\Gamma_{l} $$
(1)
$$ \frac{{\partial A\alpha_{v} \rho_{v} }}{\partial t} + \frac{{\partial A\alpha_{v} \rho_{v} v_{v} }}{\partial x} = A\Gamma_{v} $$
(2)
$$ \frac{{\partial \alpha_{l} \rho_{l} }}{\partial t} + \alpha_{l} \rho_{l} K\frac{\partial P}{{\partial t}} + \frac{{\partial \alpha_{l} \rho_{l} v_{l} }}{\partial x} + \alpha_{l} \rho_{l} v_{l} K\frac{\partial P}{{\partial x}} = \Gamma_{l} $$
(3)
$$ \frac{{\partial \alpha_{v} \rho_{v} }}{\partial t} + \alpha_{v} \rho_{v} K\frac{\partial P}{{\partial t}} + \frac{{\partial \alpha_{v} \rho_{v} v_{v} }}{\partial x} + \alpha_{v} \rho_{v} v_{v} K\frac{\partial P}{{\partial x}} = \Gamma_{v} $$
(4)
$$ \begin{aligned} \alpha_{l} \rho_{l} \frac{{\partial v_{l} }}{\partial t} & + \alpha_{l} \frac{\partial P}{{\partial x}} - P_{i} \frac{{\partial \alpha_{v} }}{\partial x} + \alpha_{l} \rho_{l} v_{l} \frac{{\partial v_{l} }}{\partial x} - VM \\ & = F_{D} + \Gamma_{l} \left( {v_{i} - v_{l} } \right) + \alpha_{l} \rho_{l} g\sin \theta - F_{{l,\,{\text{ wall }}}} \\ \end{aligned} $$
(5)
$$ \begin{aligned} \alpha_{v} \rho_{v} \frac{{\partial v_{v} }}{\partial t} & + \alpha_{v} \frac{\partial P}{{\partial x}} + P_{i} \frac{{\partial \alpha_{v} }}{\partial x} + \alpha_{v} \rho_{v} v_{v} \frac{{\partial v_{v} }}{\partial x} + VM \\ & = - F_{D} + \Gamma_{v} \left( {v_{i} - v_{v} } \right) + \alpha_{v} \rho_{v} g\sin \theta - F_{{v,\,{\text{ wall }}}} \\ \end{aligned} $$
(6)

where index l refers to the liquid phase and index v to the vapor phase. α is volume fraction, Г is mass transfer rate, \(P_{i}\) is interfacial pressure, \(F_{D}\) is inter-phasic drag force and \(F_{{\text{wall }}}\) is wall friction force.

To consider the geometric deformation of the pipe cross-section under pressure pulses, the cross-sectional area term A in Eqs. (1)–(2) is considered as a function of time and space, i.e., the elasticity of the pipe is considered. The cross-sectional area is of the form

$$ A(x, \, t) = A(x) + A_{el} [P(x, \, t)] $$
(7)

where \(A(x)\) is the nominal cross-sectional area of the pipe, and \(A_{el}\) represents the amount of change in cross-sectional area produced by a pressure pulse acting on the elastic pipe. The radial movement of the pipe wall is considered free in this paper. The study of Wylie and Streeter [13] gives the relationship between \(A_{el}\) and pressure

$$ \frac{{dA_{el} }}{A(x)} = KdP $$
(8)

where \(K = D/E\chi\), D is the tube cross-sectional diameter (m), E is the modulus of elasticity of the pipe material (N/m2), \(\chi\) is the pipe wall thickness (m). Combining Eqs. (1)–(2) and (7)–(8), the mass equation for two-phase flow in an elastic pipe can be written as

$$ \frac{{\partial \alpha_{l} \rho_{l} }}{\partial t} + \alpha_{l} \rho_{l} K\frac{\partial P}{{\partial t}} + \frac{{\partial \alpha_{l} \rho_{l} v_{l} }}{\partial x} + \alpha_{l} \rho_{l} v_{l} K\frac{\partial P}{{\partial x}} = \Gamma_{l} $$
(9)
$$ \frac{{\partial \alpha_{v} \rho_{v} }}{\partial t} + \alpha_{v} \rho_{v} K\frac{\partial P}{{\partial t}} + \frac{{\partial \alpha_{v} \rho_{v} v_{v} }}{\partial x} + \alpha_{v} \rho_{v} v_{v} K\frac{\partial P}{{\partial x}} = \Gamma_{v} $$
(10)

The Eqs. (3)–(6),(9)–(10) are still not closed equations and the unknown variables in them need to be determined by supplementary equations.

2.2 Closure Models for Control Equations

2.2.1 Equation of State

There is a differential relationship between the density of the workpiece and the pressure, internal energy and temperature as follows

$$ d\rho_{k} = \left( {\frac{{\partial \rho_{k} }}{\partial P}} \right)_{{u_{k} }} dP + \left( {\frac{{\partial \rho_{k} }}{\partial u}} \right)_{P} du_{k} $$
(11)

where the subscript k represents the vapor phase (v) or liquid phase (l). The thermodynamic coefficients involved in the above equation are constant volume compressibility \(k_{k}\), constant pressure specific heat capacity \(C_{Pk}\), constant pressure volume thermal expansion coefficient \(\beta_{k}\). Differential form written as

$$ k_{k} = - \frac{1}{{\forall_{k} }}\left( {\frac{{\partial \forall_{k} }}{\partial P}} \right)_{{T_{k} }} $$
(12)
$$ C_{Pk} = \left( {\frac{{\partial h_{k} }}{{\partial T_{k} }}} \right)_{P} $$
(13)
$$ \beta_{k} = \frac{1}{{\forall_{k} }}\left( {\frac{{\partial \forall_{k} }}{{\partial T_{k} }}} \right)_{P} $$
(14)

where \(\forall_{k} = 1/\rho_{k}\). The partial differential term of density with respect to pressure and internal energy can be further written as

$$ \left( {\frac{{\partial \rho_{k} }}{{\partial u_{k} }}} \right)_{P} = \frac{{\beta_{k} }}{{\left( {C_{Pk} - \forall_{k} \beta_{k} \rho_{k} } \right)\forall_{k} }} $$
(15)
$$ \left( {\frac{{\partial \rho_{k} }}{\partial P}} \right)_{{u_{k} }} = \frac{{C_{Pk} k_{k} - T_{k} \beta_{k}^{2} \forall_{k} }}{{\left( {C_{Pk} - \forall_{k} \beta_{k} P} \right)\forall_{k} }} $$
(16)

2.2.2 Two-Phase Flow Regime Criteria

Two-phase flow at different flow rates and vapor-liquid ratios exhibit different inter-phase interactions and interactions between the fluid and the pipe wall. The two characteristic factors, vapor volume fraction \(\alpha_{v}\) and relative velocity \(v_{r} = v_{v} - v_{l}\) are selected to classify the two-phase flow regime, as shown in Fig. 1. The flow is divided into five states, including dispersed bubbly flow, dispersed droplet flow, horizontal stratified flow and transition flow, covering the main flow characteristics of horizontal pipeline two-phase flow.

Fig. 1.
figure 1

Two-phase flow regime diagram

Full size image

When the relative velocity is low, the vapor-liquid two phases exhibit horizontal stratified flow regardless of the vapor volume fraction. When the relative velocity is higher than the critical velocity, the velocity difference between the two phases is larger and shows the characteristics of dispersed flow; at a lower vapor volume fraction, the liquid phase dominates and the flow process is diffusive bubble flow; at a higher vapor volume fraction, the vapor phase dominates and the flow process is diffusive droplet flow. There is a mutual transformation between these flow types.

The flow regime parameter R is also one of the indicators describing the flow characteristics, reflecting the degree of stratification and dispersion of the two-phase flow, proposed and validated by Tiselj [15], and contains five factors

$$ R = R_{K - H} R_{\theta } R_{\rho v} R_{v} R_{\alpha } $$
(17)

where \(R_{K - H} 1\) is Kelvin–Helmholtz instability factor, which is assumed to be the primary reason for the transition of flow regime in CIWH phenomenon.

2.2.3 Virtual Mass

During the flow process, forces exist between the phases due to the spatial and temporal variation of the relative velocities of the two phases. For the part of the interphase force caused by the acceleration of the vapor or liquid phase, the virtual mass is used to characterize it. The virtual mass of a solid sphere (bubble or droplet) in an ideal fluid has a theoretical analytical solution, but the form of the virtual mass term in a real two-phase flow is not known. In the WAHA3 method, the virtual mass term in the momentum equation takes the form given by Drew [16]

$$ VM = C_{VM} \left( {\frac{{\partial v_{v} }}{\partial t} + v_{l} \frac{{\partial v_{v} }}{\partial x} - \frac{{\partial v_{l} }}{\partial t} - v_{v} \frac{{\partial v_{l} }}{\partial x}} \right) $$
(18)

where \(C_{VM}\) is the virtual mass coefficient, which is determined by the flow regime parameters, mixture density, and vapor volume fraction

$$ C_{VM} = \left\{ {\begin{array}{*{20}c} {C_{1} \frac{{1 + 2\alpha_{v} }}{{2\alpha_{l} }}} & {\alpha_{v} \le 0.4} \\ {C_{1} \left[ {1.5 - 10\left( {\alpha_{v} - 0.4} \right)\left( {\alpha_{v} - 0.6} \right)} \right]} & {0.4 < \alpha_{v} < 0.6} \\ {C_{1} \sqrt {\left( {\frac{{3 - 2\alpha_{v} }}{{2\alpha_{v} }}} \right)^{2} + \frac{{\alpha_{l} \left( {2\alpha_{v} - 2} \right)}}{{\left( {\alpha_{l} + \frac{{\alpha_{l} \rho_{v} }}{{\rho_{l} }}} \right)^{2} }}} } & {\alpha_{v} \ge 0.6} \\ \end{array} } \right. $$
(19)

2.2.4 Heat Transfer Correlations

The heat transfer at the phase interface in the two-phase energy equation is

$$ Q_{ik} = H_{ik} \left( {T_{s} - T_{k} } \right) $$
(20)

where \(T_{s}\) is the saturation temperature, which is only related to the work substance and pressure. \(H_{ik}\) is the volumetric interfacial heat transfer coefficient, which is calculated by the method proposed by Brucker [14].

3 Numerical Solution Method

3.1 Splitting Operator Method

The basic governing equations of mass, momentum and energy for individual phases can be transformed to the following matrix form

$$ {\varvec{A}}\frac{\partial \varPhi }{{\partial t}} + {\varvec{B}}\frac{\partial \varPhi }{{\partial x}} = {\varvec{S}} $$
(21)

where \(\phi\) represents a vector of the variables for solving \(\phi = \left( {P\alpha_{v} v_{l} v_{v} u_{l} u_{v} } \right)^{\rm{ \top }}\) and A, B are 6-times-6 matrices and S is the source vector. Depending on the time scale of the source term, the source is split into a non-relaxation source and a relaxation source

$$ {\varvec{S}} = {\varvec{S}}_{NR} + {\varvec{S}}_{R} $$
(22)

Non-relaxation source, which contains friction and volume force terms, is closely related to the convective term of the model. The relaxation source, which establishes the heat and mechanical equilibrium between two phases, reflects the heat, mass and momentum transfer between the phases.

$$ {\varvec{A}} = \left( {\begin{array}{*{20}c} {G_{l} \left( {\frac{{\partial \rho_{l} }}{\partial P}} \right)_{u} + \rho_{l} K} & { - \frac{{\rho_{l} }}{{\alpha_{l} }}} & 0 & 0 & 0 & 0 \\ {G_{v} \left( {\frac{{\partial \rho_{v} }}{\partial P}} \right)_{u} + \rho_{k} K} & {\frac{{\rho_{v} }}{{\alpha_{v} }}} & 0 & 0 & 0 & 0 \\ 0 & 0 & {\rho_{l} + \frac{{C_{VM} }}{{\alpha_{l} }}} & { - \frac{{C_{VM} }}{{\alpha_{l} }}} & 0 & 0 \\ 0 & 0 & { - \frac{{C_{VM} }}{{\alpha_{v} }}} & {\rho_{v} + \frac{{C_{VM} }}{{\alpha_{v} }}} & 0 & 0 \\ {PK} & { - \frac{P}{{\alpha_{l} }}} & 0 & 0 & {\rho_{l} } & 0 \\ {PK} & {\frac{P}{{\alpha_{v} }}} & 0 & 0 & 0 & {\rho_{v} } \\ \end{array} } \right)G_{k} = \frac{1}{{1 - \frac{P}{{\rho_{k}^{2} }}\left( {\frac{{\partial \rho_{k} }}{{\partial u_{k} }}} \right)_{P} }} $$
$$ B\,{ = }\left( {\begin{array}{*{20}l} {v_{l} \left[ {G_{l} \left( {\frac{{\partial \rho_{l} }}{\partial P}} \right)_{u} + \rho_{l} K} \right]} \hfill & { - \frac{{\rho_{l} v_{l} }}{{\alpha_{l} }}} \hfill & {\rho_{l} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {v_{v} \left[ {G_{v} \left( {\frac{{\partial \rho_{v} }}{\partial P}} \right)_{u} + \rho_{k} K} \right]} \hfill & {\frac{{\rho_{v} v_{v} }}{{\alpha_{v} }}} \hfill & 0 \hfill & {\rho_{v} } \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & { - \frac{{P_{i} }}{{\alpha_{l} }}} \hfill & {\rho_{l} v_{l} + \frac{{C_{VM} }}{{\alpha_{l} }}v_{v} } \hfill & { - \frac{{C_{VM} }}{{\alpha_{l} }}v_{l} } \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & {\frac{{P_{i} }}{{\alpha_{v} }}} \hfill & { - \frac{{C_{VM} }}{{\alpha_{v} }}v_{v} } \hfill & {\rho_{v} v_{v} + \frac{{C_{VM} }}{{\alpha_{v} }}v_{l} } \hfill & 0 \hfill & 0 \hfill \\ {Pv_{l} K} \hfill & { - \frac{{Pv_{l} }}{{\alpha_{l} }}} \hfill & P \hfill & 0 \hfill & {\rho_{l} v_{l} } \hfill & 0 \hfill \\ {Pv_{v} K} \hfill & {\frac{{Pv_{v} }}{{\alpha_{v} }}} \hfill & 0 \hfill & P \hfill & 0 \hfill & {\rho_{v} v_{v} } \hfill \\ \end{array} } \right) $$
$$ {\varvec{S}}_{R} = \left( {\begin{array}{*{20}c} {\frac{{G_{l} }}{{\alpha_{l} }}\left\{ {\Gamma_{l} - \left[ {Q_{il} + \Gamma_{l} \left( {h_{l} - u_{l} } \right)} \right]\frac{1}{{\rho_{l} }}\left( {\frac{{\partial \rho_{l} }}{{\partial u_{l} }}} \right)_{P} } \right\}} \\ {\frac{{G_{v} }}{{\alpha_{v} }}\left\{ {\Gamma_{v} - \left[ {Q_{iv} + \Gamma_{v} \left( {h_{v} - u_{v} } \right)} \right]\frac{1}{{\rho_{v} }}\left( {\frac{{\partial \rho_{v} }}{{\partial u_{v} }}} \right)_{P} } \right\}} \\ {\frac{1}{{\alpha_{l} }}F_{D} + \frac{1}{{\alpha_{l} }}\Gamma_{l} \left( {v_{i} - v_{l} } \right) + \rho_{l} g\sin \theta } \\ { - \frac{1}{{\alpha_{v} }}F_{D} + \frac{1}{{\alpha_{v} }}\Gamma_{v} \left( {v_{i} - v_{v} } \right) + \rho_{v} g\sin \theta } \\ {\frac{1}{{\alpha_{l} }}\left[ {Q_{il} + \Gamma_{l} \left( {h_{l} - u_{l} } \right)} \right]} \\ {\frac{1}{{\alpha_{v} }}\left[ {Q_{iv} + \Gamma_{v} \left( {h_{v} - u_{v} } \right)} \right]} \\ \end{array} } \right){\varvec{S}}_{NR} = \left( {\begin{array}{*{20}c} { - v_{l} F_{{l,{\text{ wall }}}} \frac{{G_{l} }}{{\rho_{l} \alpha_{l} }}\left( {\frac{{\partial \rho_{l} }}{{\partial u_{l} }}} \right)_{P} } \\ { - v_{v} F_{{v,{\text{ wall }}}} \frac{{G_{v} }}{{\rho_{v} \alpha_{v} }}\left( {\frac{{\partial \rho_{v} }}{{\partial u_{v} }}} \right)_{P} } \\ { - \frac{1}{{\alpha_{l} }}F_{{l,{\text{ wall }}}} } \\ { - \frac{1}{{\alpha_{v} }}F_{{v,{\text{ wall }}}} } \\ {\frac{1}{{\alpha_{l} }}v_{l} F_{{l,{\text{ wall }}}} } \\ {\frac{1}{{\alpha_{v} }}v_{v} F_{{v,{\text{ wall }}}} } \\ \end{array} } \right) $$

3.2 Two-Step Iterative Algorithm

The final model with all the control and closure equations is

$$ {\varvec{A}}\frac{\partial \varPhi }{{\partial t}} + {\varvec{B}}\frac{\partial \varPhi }{{\partial x}} = {\varvec{S}}_{NR} + {\varvec{S}}_{R} $$
(23)

After applying the operator splitting method, the solution of \(\varPhi\) is iterated in two steps, as shown in Fig. 2.

Fig. 2.
figure 2

Schematic of the two-step iterative method

Full size image

The superscript n in \(\varPhi_{j}^{(n)}\) indicates the time node where \(\varPhi\) is currently located, and the subscript j indicates the spatial node where \(\varPhi\) is currently located. The iteration of a time step is split into two parts, and the variable \(\varPhi_{j}^{(n)}\) at a certain spatial position needs to experience an intermediate time *(n) when crossing from time (n) to time (n + 1).

3.2.1 Convective Iterative

Only consider the effect of non-relaxation sources in the convection step, Eq. (23) can be re-written as

$$ \frac{\partial \varPhi }{{\partial t}} + {\varvec{C}}\frac{\partial \varPhi }{{\partial x}} = {\varvec{A}}^{ - 1} {\varvec{S}}_{NR} $$
(24)

where the Jacobain matrix \(C = A^{ - 1} B\) can be diagonalized in terms of its eigen vector V and eigen value matrix \({{\varvec{\Lambda}}}\) as follows

$$ {\varvec{C}} = {\varvec{V}}{{\varvec{\Lambda}}}{\varvec{V}}^{ - 1} $$
(25)

Applying the diagonalization result to Eq. (23) and setting s \(R_{F} = - V^{ - 1} S_{NR}\) yields

$$ {\varvec{V}}^{ - 1} \left( {\frac{\partial \varPhi }{{\partial t}} + {\varvec{V}}{{\varvec{\Lambda}}}{\varvec{V}}^{ - 1} \frac{\partial \varPhi }{{\partial x}} + {\varvec{R}}_{F} } \right) = 0 $$
(26)

Introduce the characteristic variable \({\varvec{W}}_{6 \times 1}\), which is related to the original variable \(\varPhi\) as

$$ \partial {\varvec{W}} = {\varvec{V}}^{ - 1} \partial \varPhi + {{\varvec{\Lambda}}}^{ - 1} {\varvec{V}}^{ - 1} {\varvec{R}}_{F} \partial x $$
(27)

Equation (26) can be expressed by the characteristic variables

$$ \frac{{\partial {\varvec{W}}}}{\partial t} + {{\varvec{\Lambda}}}\frac{{\partial {\varvec{W}}}}{\partial x} = 0 $$
(28)

The TVD [17] (Total Variation Diminishing) format with restriction factors is used to differential Eq. (28) in the space of eigenvariables

$$ \frac{{{\varvec{W}}_{j}^{*(n)} - {\varvec{W}}_{j}^{(n)} }}{{\Delta t_{NR} }} + {{\varvec{\Lambda}}}^{ + } \frac{{{\varvec{W}}_{j}^{(n)} - {\varvec{W}}_{j - 1}^{(n)} }}{\Delta x} + {{\varvec{\Lambda}}}^{ - } \frac{{{\varvec{W}}_{j + 1}^{(n)} - {\varvec{W}}_{j}^{(n)} }}{\Delta x} = 0 $$
(29)

where the eigenvalue diagonal array in Eq. (29) splits into two parts, and the split \({{\varvec{\Lambda}}}\) is still a diagonal array, and its relationship in spatial location is shown in Fig. 3.

$$ {{\varvec{\Lambda}}}_{j}^{(n)} = {{\varvec{\Lambda}}}_{j - 1/2}^{(n)} + {{\varvec{\Lambda}}}_{j + 1/2}^{(n)} = {{\varvec{\Lambda}}}^{ + } + {{\varvec{\Lambda}}}^{ - } $$
(30)
Fig. 3.
figure 3

Spatial discrete schematic

Full size image

Replacing the characteristic variables with the original variables \(\varPhi\), \(\varPhi_{j}^{*(n)}\) can be deduced from Eq. (28) as

$$ \begin{gathered} \varPhi_{j}^{*(n)} = \varPhi_{j}^{(n)} - \Delta t_{NR} \cdot \left( {{\varvec{V}}{{\varvec{\Lambda}}}^{ + } {{\varvec{\Lambda}}}^{ - 1} {\varvec{V}}^{ - 1} {\varvec{R}}_{F} + {\varvec{V}}{{\varvec{\Lambda}}}^{ - } {{\varvec{\Lambda}}}^{ - 1} {\varvec{V}}^{ - 1} {\varvec{R}}_{F} } \right. \hfill \\ \left. { +\, {\varvec{V}}{{\varvec{\Lambda}}}^{ + } {\varvec{V}}^{ - 1} \frac{{\varPhi_{j}^{(n)} - \varPhi_{j - 1}^{(n)} }}{\Delta x} + {\varvec{V}}{{\varvec{\Lambda}}}^{ - } {\varvec{V}}^{ - 1} \frac{{\varPhi_{j + 1}^{(n)} - \varPhi_{j}^{(n)} }}{\Delta x}} \right) \hfill \\ \end{gathered} $$
(31)

The time step size \(\Delta t_{NR}\) is evaluated based on the CFL [18] (Courant-Friedrichs-Levy) criterion as follows

$$ \Delta t_{NR} \le \frac{\Delta x}{{\max \left| {\lambda_{t} } \right|}} $$
(32)

3.2.2 Relaxation Iterative

In the relaxation step only consider the relaxation source. The velocity terms in the variables (corresponding to the momentum equation) are calculated from the assumptions on the relative and mixed velocities. In the calculation of the other variables, momentum transfer is not considered, but only the effects of heat and mass transfer at the phase interface.

$$ {\varvec{A}}_{T} \frac{\partial \psi }{{\partial t}} = {\varvec{S}}_{R - HMT} $$
(33)
$$ {\varvec{A}}_{T} = \left( {\begin{array}{*{20}c} {\left( {\frac{{\partial \rho_{l} }}{\partial P}} \right)_{{T_{l} }} + \rho_{l} K} & { - \frac{{\rho_{l} }}{{\alpha_{l} }}} & {\left( {\frac{{\partial \rho_{l} }}{{\partial T_{l} }}} \right)_{P} } & 0 \\ {\left( {\frac{{\partial \rho_{v} }}{\partial P}} \right)_{{T_{v} }} + \rho_{v} K} & {\frac{{\rho_{v} }}{{\alpha_{v} }}} & 0 & {\left( {\frac{{\partial \rho_{v} }}{{\partial T_{v} }}} \right)_{P} } \\ {PK + \rho_{l} \left( {\frac{{\partial u_{l} }}{\partial P}} \right)_{{T_{l} }} } & { - \frac{P}{{\alpha_{l} }}} & {\rho_{l} \left( {\frac{{\partial u_{l} }}{{\partial T_{l} }}} \right)_{P} } & 0 \\ {PK + \rho_{v} \left( {\frac{{\partial u_{v} }}{\partial P}} \right)_{{T_{v} }} } & {\frac{P}{{\alpha_{v} }}} & 0 & {\rho_{v} \left( {\frac{{\partial u_{v} }}{{\partial T_{v} }}} \right)_{P} } \\ \end{array} } \right) $$
$$ {\varvec{S}}_{R - HMT} = \left( {\begin{array}{*{20}c} {\frac{{G_{l} }}{{\alpha_{l} }}\left\{ {\Gamma_{l} - \left[ {Q_{il} + \Gamma_{l} \left( {h_{l} - u_{l} } \right)} \right]\frac{1}{{\rho_{l} }}\left( {\frac{{\partial \rho_{l} }}{{\partial u_{l} }}} \right)_{P} } \right\}} \\ {\frac{{G_{v} }}{{\alpha_{v} }}\left\{ {\Gamma_{v} - \left[ {Q_{iv} + \Gamma_{v} \left( {h_{v} - u_{v} } \right)} \right]\frac{1}{{\rho_{v} }}\left( {\frac{{\partial \rho_{v} }}{{\partial u_{v} }}} \right)_{P} } \right\}} \\ {\frac{1}{{\alpha_{l} }}\left[ {Q_{il} + \Gamma_{l} \left( {h_{l} - u_{l} } \right)} \right]} \\ {\frac{1}{{\alpha_{v} }}\left[ {Q_{iv} + \Gamma_{v} \left( {h_{v} - u_{v} } \right)} \right]} \\ \end{array} } \right)\psi = \left( {\begin{array}{*{20}c} P \\ {\alpha_{v} } \\ {T_{l} } \\ {T_{v} } \\ \end{array} } \right) $$

The time step of the relaxation iterative is related to the relative rate of change of the variable \(\psi\)

$$ \frac{\partial \psi }{{\partial t}} = {\varvec{A}}_{T}^{ - 1} {\mathbf{S}}_{R - HMT} = \left( {\begin{array}{*{20}l} {\dot{P}} \hfill & {\dot{\alpha }_{v} } \hfill & {\dot{T}_{l} } \hfill & {\dot{T}_{v} } \hfill \\ \end{array} } \right)^{\rm{ \top }} $$
(34)

The relaxation time step is evaluated as follows

$$ \Delta t_{R} = 0.01\min \left( {t_{1} ,t_{2} ,t_{3} ,t_{4} } \right) $$
(35)
$$ \left\{ {\begin{array}{*{20}l} {t_{1} = \frac{P}{{\max ( \in ,|\dot{P}|)}}} \hfill \\ {t_{2} = \frac{{\max \left( {0.01,\min \left( {\alpha _{v} ,\alpha _{l} } \right)} \right)}}{{\max \left( { \in ,\left| {\dot{\alpha }_{v} } \right|} \right)}}} \hfill \\ {t_{3} = \frac{{\max \left( {\left| {T_{l} - T_{s} } \right|} \right)}}{{\max \left( { \in ,\left| {\dot{T}_{l} } \right|} \right)}}} \hfill \\ {t_{4} = \frac{{\max \left( {\left| {T_{v} - T_{s} } \right|} \right)}}{{\max \left( { \in ,\left| {\dot{T}_{v} } \right|} \right)}}} \hfill \\ \end{array} } \right. $$
(36)

The relative velocity is iterated in the relaxation step as

$$ v_{r}^{(n + 1)} = \frac{{v_{r}^{*(n)} }}{{1 - \Delta t_{R} S_{vr} }} $$
(37)

where

$$ S_{vr} = \left\{ {\begin{array}{*{20}l} {\frac{{ - C_{i} \left| {v_{r} } \right|\rho_{m} + \alpha_{v} \rho_{v} \Gamma_{v} }}{{\alpha_{v} \alpha_{l} \rho_{l} \rho_{v} + C_{VM} \rho_{m} }}} \hfill & {\Gamma_{v} < 0} \hfill \\ {\frac{{ - C_{i} \left| {v_{r} } \right|\rho_{m} - \alpha_{l} \rho_{l} \Gamma_{v} }}{{\alpha_{v} \alpha_{l} \rho_{l} \rho_{v} + C_{VM} \rho_{m} }}} \hfill & {\Gamma_{v} > 0} \hfill \\ \end{array} } \right. $$
(38)

4 Results and Discussions

4.1 Verification with Benchmark Problem (Two-Phase Shock Tube Case)

The one-dimensional Riemann two-phase shock wave tube problem is a typical benchmark problem for condensation-induced water hammer, reflecting the ability of the model to capture discontinuous physical states and millisecond pressure transients.

A straight tube of length 1 m is filled with vapor-liquid mixture, the diameter of the tube is 19 mm, the wall thickness is 1.6 mm, and the modulus of elasticity of the tube material is 760 MPa, as shown in Fig. 4. In the initial state, a thin film at the center of the tube divides the tube into two parts, both with an initial vapor volume fraction of 0.3. The densities of the left and right side of the mixture are 998.64 and 998.23 kg/m3.

Fig. 4.
figure 4

Schematic of the initial state of the shock wave tube

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Fig. 5.
figure 5

Numerical solution of the Riemann shock wave tube problem for pressure in elastic and rigid tubes (at 0.42 ms)

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The test starts after the film is completely ruptured and the vapor-liquid mixture diffuses from high to low pressure, forming a surge inside the tube. The surge tube is discretized into 100 nodes, and the calculated results are shown in Fig. 5, reflecting the pressure distribution inside the tube at 0.42 ms after the vapor-liquid mixture with different pressure starts to contact. The calculated results show good agreement with the analytical values.

4.2 Verification with Experimental Results of PMK-2

In order to study the CIWH phenomenon that occurs during the entry of cold water into steam-filled pipes in the water circuit of a nuclear power facility, the Hungarian KFKI Institute constructed a model steam pipe and conducted the PMK-2 test.

The test section of the PMK-2 test is a tube with a length-to-diameter ratio of about 39. The pipe is filled with saturated water vapor at a pressure of 1.45 MPa at the initial moment, and the saturation temperature of the water vapor at this pressure is 470 K. The pipe and boundary conditions can be simplified as shown in Fig. 6. The volume of the steam kettle at the end is sufficiently large that the outlet pressure can be considered to be approximately constant.

Fig. 6.
figure 6

PMK-2 test section tube schematic

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The numerical solution method of this paper is used to calculate the PMK-2 test conditions, and the pipeline is divided into a total of 10 intervals (11 nodes), and the occurrence of pressure pulses is successfully captured. A comparison of the calculation results in this paper with the PMK-2 test results is shown in Fig. 7.

Fig. 7.
figure 7

Comparison of pressure time history between PMK-2 test and the results calculated in this paper

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In terms of peak pressure, the pressure pulse amplitude of 17.1 MPa obtained by the calculation method in this paper, compared with the highest value of 17.4 MPa recorded in the test, the relative error is 1.7%, indicating that the calculation method can be more accurate in forecasting the impact pressure when water hammer occurs under the test conditions.

4.3 Analysis of the Effect of Inlet Velocity on CIWH

The temperature and pressure parameters of the PMK-2 test are applied, and the injection velocity of subcooled water is varied (0.1 m/s–0.4 m/s) to investigate the effect of subcooled water flow rate on the CIWH phenomenon in the tube. A total of five velocities are numerically calculated for the model with the modulus of elasticity of the pipe material set to 206 GPa, the model discretization are 10 nodes, the velocity boundary condition at the left inlet and the pressure boundary condition at the right outlet (Figs. 8, 9 and Table 1).

Table 1. Parameters used in the analysis of inlet velocity
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Fig. 8.
figure 8

Effect of subcooled water inlet velocity on CIWH

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Fig. 9.
figure 9

Effect of subcooled water inlet velocity on peak pressure and time of CIWH occurrence

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With the increasing flow rate of subcooled water (0.1 m/s to 0.4 m/s), the peak pressure of condensate water hammer is growing, but the growth rate is gradually slowing down; the occurrence time of water hammer is constantly in advance, but when the flow rate is greater than 0.3 m/s, the change of the occurrence time is no longer obvious and is maintained at 0.5 s or less.

4.4 Analysis of the Effect of Inlet Water Temperature on CIWH

Following the calculation conditions in Sect. 4.3, the injection velocity of subcooled water remains unchanged at 0.24 m/s, and only the liquid phase temperature is changed from 285 K to 325 K, every 10 K for a grade, for a total of 5 water temperature conditions (Figs. 10 and 11).

Fig. 10.
figure 10

Effect of subcooled water temperature on CIWH

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Fig. 11.
figure 11

Effect of subcooled water temperature on peak pressure and time of CIWH occurrence

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Except for the liquid phase temperature of 325 K, the sudden pressure change occurs about 2 s after the cold water starting inject. The condensate water hammer pressure peak is the largest at 285 K, reaching 18.5 MPa, and the lowest at 315 K, 7.0 MPa, and enter a state of continuous oscillation after the water hammer occurs. The obvious pressure pulse phenomenon is no longer observed when the temperature increased to 325 K.

4.5 Analysis of the Effect of Elastic Modulus on CIWH

Using the numerical algorithm of this paper, a computational study of CIWH phenomenon occurring in pipes with different modulus of elasticity (E) is done. The injection velocity and temperature of the supercooled water are kept constant at 0.24 m/s and 295 K.

Fig. 12.
figure 12

Effect of tube elasticity modulus on CIWH

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The change in the elasticity of the pipe structure does not have a significant effect on the time of occurrence of CIWH, and the peak pressure occurs within 2.1 ± 0.05 s for several conditions in Fig. 12. The change of the pressure peak is mainly related to the deformation of the pipe cross-section. In the pipe with greater stiffness, the smaller the growth of the cross-sectional area caused by the pressure pulse, the smaller the cavitation area formed after gas condensation, and the smaller the stroke of the bubble collapse effect, which shows a lower pressure peak.

Fig. 13.
figure 13

Comparison of changes in tube cross-sectional area before and after the occurrence of CIWH (E = 760 MPa)

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Fig. 14.
figure 14

Comparison of changes in tube cross-sectional area before and after the occurrence of CIWH (E = 206 GPa)

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As shown in Fig. 13 and Fig. 14, comparing the cross-sectional area before and after the water hammer, we can see the extent of the deformation of the pipe caused by the pressure pulse. For the pipe with E = 206 GPa, although the deformation of the pipe cross-section is much smaller than the former, the pressure pulse has a broader impact on the pipe deformation.

4.6 Analysis of the Effect of Tube Diameter on CIWH

The tube diameter is directly related to the mass flow rate of subcooled water injection and the contact area between the two phases, and also has an impact on the peak pressure and occurrence time of condensate hammer. The numerical model of this paper is used to calculate the pressure time history of the same temperature of subcooled water with the same flow rate into different diameter pipes, the modulus of elasticity is taken as 206 GPa, and the diameters are taken as 63 mm, 68 mm, 70.5 mm, 73 mm, 75.5 mm, 78 mm and 80.5 mm.

As shown in Fig. 15, in the diameter range of 70.5 to 80.5 mm, with the increase of pipe diameter, the peak pressure of condensate hammer increases up to 20.3 MPa, but the time of pressure pulse occurs backward, the peak pressure occurs at 1.96 s under 70.5 mm pipe diameter, and this moment is delayed to 2.44 s under 80.5 mm pipe diameter. When the pipe diameter is less than 70.5 mm, the peak pressure will be When the pipe diameter is less than 70.5 mm, the pressure peak will be much smaller than the result recorded in PMK-2 test.

Fig. 15.
figure 15

Effect of tube diameter on CIWH

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5 Conclusions

The computational results of the Riemann two-phase surge problem confirm the accuracy and sensitivity of the computational method in this thesis, demonstrating that the compressible two-fluid six-equation control model is capable of capturing pressure and velocity transients on the millisecond scale. Numerical simulations of the important experimental PMK-2 for condensation-induced water hammer are also performed on top of the benchmark problem, and a good match is achieved in terms of pressure pulse amplitude prediction and location of occurrence.

Based on the test equipment conditions, this paper also calculates the subcooled water flow rate and subcooled water temperature factors, which are of interest in this research area. The results show that as the inlet flow rate increases, the peak pressure pulse rises and CIWH occurs earlier; as the subcooled water temperature increases, the peak pressure falls and no longer triggers CIWH if the subcooled water temperature decreases to a certain extent. The pressure peak of CIWH is largely affected by the tube diameter. When the pipe cross-sectional area increases by 7.2% from an initial diameter of 70.5 mm, the pressure peak increases up to 2.1 times if the original, while CIWH is not be observed when the pipe diameter turns to be small.