Studying Dynamical and Hydraulic Characteristics of the Hydraulically Suspended Passive Shutdown Subassembly (HS-PSS) and Validating with a Prototypic Test Sample

Abstract

The pool-type demonstrative China Fast Reactor 600 (CFR-600) adopted a series of improved safety designs, among which the hydraulic suspended passive shutdown subassembly (HS-PSS) is employed for inherent safety enhancement, especially suitable against unprotected loss of flow accident (ULOF). In this article, functional requirements for HS-PSS design in hydro-dynamic aspects are proposed, with the corresponding performance indicators discussed. To address these functional requirements, qualitative analysis on the equilibrium solution properties of the nonlinear dynamical model equation of the hydraulic moving body (HMB) in the HS-PSS are conducted, which leads to the determination of an applicable design parameter domain of the HMB for its practical design from the broad range of structural and parametric design options available for HS-PSS. Furtherly, hydraulically characterizing and modeling the constituent paths and consequent fluid network, hydraulic characteristics of the HS-PSS, as well as the coupled hydro-dynamic motion behaviors of the HMB for suspension and dropping states, were simulated and test-validated. Considering the HS-PSS hydro-dynamic behaviors, key indicators such as critical flowrate, drop time, hydraulic self-tightening performance, as well as the hydraulic characteristics curve are for fulfilling the functional requirements. Meanwhile, through sensitivity study of some structural parameters‘ impact on hydraulic characteristics, some most sensible structural parameters for adjusting and optimizing detailed design are observed. The work is quite significant in supporting the conceptual design of the HS-PSS as well as its engineering improvement.

1. Introduction

As one of the six types of recommended nuclear energy systems by the Generation IV International Forum (GIF), the sodium-cooled fast reactor (SFR) exhibits notable advantages in safety, economy, efficient nuclear resource utilization, and proliferation [1,2,3]. Furthermore, SFR stands out as the reactor type with the fastest R&D progress, the most mature technique, and the development level which is the closest to meet the commercial operation requirement. Of course, development of SFR still faces many challenges, such as safety issues [4,5].

The reactor protection system (RPS) is designed in SFR, which automatically initiates shutdown of the reactor in the event of deviating from its normal operation state and posing a safety risk on the core. Anticipated transients without scram (ATWS), which refers to the unprotected transients (mainly ULOF, UTOP, and ULOHS) due to RPS failure and the inability to shutdown the reactor, represents one of the most severe accident scenarios. Although the likelihood of ATWS occurring is generally very low, its occurrence carries a significant risk of causing core disruptive accidents (CDA). Related studies indicate that core damage of SFR under ATWS condition can well be avoided by a passive introduction of quite limited negative reactivity into the core [6,7].

Among the countermeasures to prevent occurrence of CDAs in SFR, a passive shutdown system (PSS) is an important measure against ATWS to further improve core safety and serve as a redundant protection system beyond RPS [8]. Various schemes of PSS for SFR were proposed or are under development internationally, such as the gas expansion module (GEM), enhanced thermal elongation mechanisms (ETEM), self-actuating shutdown system (SASS) based on the Curie point effect, and hydraulically suspended passive shutdown subassembly (HS-PSS), etc.

The demonstrative China Fast Reactor 600 (CFR-600) is a pool-type SFR with the capability of power production of about 600 MW in electricity and 1500 MW in thermal. CFR-600 aims to demonstrate industrial technology and a closed fuel cycle. It is designed and launched in pursuit of high economy, high breeding ratio, longer lifetime, high burnup of fuel, and thermal efficiency. Meanwhile, CFR-600 is designed according to the safety requirements of the Generation-IV system [3].

In addition to the inheritance of distinguished characteristics and features of the China Experimental Fast Reactor (CEFR) as a pool-type fast reactor, CFR-600 incorporates a series of improved safety designs, including equipping the passive shutdown system (PSS) to enhance inherent safety. The pool-type CFR-600 is proved to have good ability to resist UTOP accidents, in the development of PSS for CFR-600, therefore, priority should be given to passive shutdown technologies that are capable of coping with more severe ULOF accidents [9]. Based on this consideration, as well as considering other factors, such as its relative technical maturity and reliability over other schemes, etc., CFR-600 selects to be equipped with HP-PSS, which is a PSS sensitive to coolant flowrate during loss of flow accident.

Among the current major fast reactor countries, only Russia and China adopt HS-PSS schemes. Since 1988, Russia has conducted a series of research on it and accumulated operational experience, while research on HS-PSS started relatively late in China.

Since 1988, IPPE of Russia (former Soviet Union) has been conducting research on passive shutdown devices beyond the normal shutdown system of SFRs in order to further enhance safety. Progress was made from 1990 to 1995 [7]. In 1988–1989, IPPE designed and manufactured two prototypic hydraulic suspended rod (HSR) test samples based on BR-10 standard assembly. A series of tests on the HSRs were conducted within water and in a BR-10 reactor. Through the tests, the various hydro-dynamic performance parameters of HSR were verified, and thus reference standards for HSR of the BR-10 reactor were confirmed.

In 1989, IPPE began the R&D of the HSR for the BN-600 reactor. Full-size prototypic test samples of HSR for water testing were designed and manufactured. Hydraulic tests were conducted on the HSR samples of varying design dimensions, ultimately leading to the recommendation of an optimized design scheme for the BN-600 reactor. Furtherly, IPPE also developed prototypic HSR test samples for the BN-800 reactor and carried out dropping tests in water [10], verifying the hydraulic performance and structural reliability of the HSR.

In China, due to its own structural form and in-core conditions in CFR600, independent and original development of the passive shutdown system in CFR-600 is led by CIAE, in collaboration with other relevant R&D units. Design, research, manufacturing, experimental verification, and test validation are carried out simultaneously and iteratively. In developing HS-PSS, aspects under operation conditions such as functional requirements; hydro-dynamic performances of HS-PSS, negative reactivity insertion effect; mechanical properties and materials processing; structural vibration, flow-induced vibration (FIV), and seismic characteristics; in-core irradiation; reliability, etc., are evaluated and validated. Among them, HS-PSS hydro-dynamic behaviors corresponding to its diverse structures, forces, and movements are most closely related to the implementation of its functions. They constitute the most primary and important research content for HS-PSS design.

In 2014, Hu et al. [9] conducted study on the passive shutdown technology for pool-type SFR. They proposed that China’s demonstration fast reactor should prioritize the development of hydraulic suspension passive shutdown mechanisms. Wang et al. [11] calculated the pressure drop and flow distribution of the hydraulic movable bodies (HMBs) during its movement within HS-PSS based on the fluid network model, from which the HMB dropping curve as well as the drop time were estimated. Peng et al. [12] developed a one-dimensional computational program based on the model for predicting the HMB drop process within the HS-PSS. Yuan et al. [13] conducted experimental research with some typical components, component combinations, as well as a full-length test piece of the preliminary designed HS-PSS. Hydraulic characteristics of the components and of the component combinations for HS-PSS were investigated, and the HS-PSS flow resistance model was modified. Wang et al. [14,15] analyzed the mathematical model and hydraulic characteristics of the HMB dropping process under different flow conditions of HS-PSS operation. In the above-mentioned modeling and analysis, relevant work, experiences, and progress in similar situations or applications, such as those in [16,17,18,19,20,21], etc., were referred. A hydro-dynamic design and prediction code was developed and sensitivity analysis of various factors affecting the performance of HS-PSS were conducted for conceptual design and continuous improving. Meanwhile and from then on, experimental investigation and verification of the hydraulic characteristics of the HS-PSS prototype test samples, as well as their function and performance validation tests, are continuously carried out. In addition, Liu [22], Yang et al. [23] simulated the HMB dropping motion within the HS-PSS through CFD, and concluded that the gap width and buffer structure were the most important factors affecting the drop velocity. Furtherly, Wang et al. [24] completes an experimental and numerical research on FIV characteristics with the HS-PSS test sample. These all contribute to providing support for the design and development of HS-PSS for CFR-600.

In this article, qualitative analysis on the global dynamics of HMB movement is conducted, referring to the working principle, structural feature, and functional requirements of the HS-PSS, so as to select and define applicable parametric domain for its functioning. Detailed study on the working performances of the HS-PSS by integrating HMB dynamics and hydraulics, along with code simulations and test validation with a subassembly test sample is reported, which aims to support both the HS-PSS conceptual design and hydraulic optimization for CFR-600.

2. Conceptual Mechanism and Functional Requirements for Design of HS-PSS

HS-PSS is mainly composed of a cylindrical outer sleeve and an internal movable hydraulic moving body (HMB). The neutron absorbers in the form of a rod bundle are fixedly installed in HMB as part of it, introducing negative reactivity to the core when HMB inserts downwards into the core and permits coolant, during HMB dropping, to flow upwards through flow paths within HMB and the annular gap between the outer sleeve and HMB. As the major part of redundancy and supplement to RPS, HS-PSS, with its complex structure and specific functional design, can ensure reactor shutdown in the most severe unprotected loss of flow (ULOF) accident while RPS fails. Dynamic behavior as well as the related functional implementation of HS-PSS are closely connected to both the normal operation and safety characteristics of the reactor. Figure 1 illustrates schematically the basic structure of HS-PSS.

As seen in Figure 1, coolant flows upward through the core, entering HS-PSS from below and sweeping HMB from both inside and outside. Part of the coolant flow penetrates upward through the inner of HMB to the cool absorber, then discharges from its outlet holes, merging with the flow from the outside annular gap. Merged flow ultimately exits the core from the top exit of HS-PSS.

Safety function of HS-PSS is realized by harnessing the distinct motion of the absorber-containing HBM within the outer sleeve in response to coolant flow variations. During normal operation with a high enough coolant flowrate, the HMB is subject to substantial hydraulic thrust and hence remains staying at the upper working position (UWP) under the constraint of the upper limit device. In the event of a ULOF accident accompanying decreasing of coolant flow, the hydraulic thrust on the HMB decreases and the HMB drops to the lower working position (LWP). Moving with HMB, the absorber is inserted into the core region, thereby passively triggering shutdown.

In order to ensure effective response to ULOF without affecting the reactor in normal operation, the HMB in HS-PSS should be designed to appropriately act and swiftly respond to variations in coolant flowrate following accidents. Meanwhile, both unintentional HMB dropping in normal operation and delayed or slow dropping in ULOF should be avoided. That is, static and dynamical features as well as the related parameters of HMB, such as suspension hydraulic characteristics, critical flowrate triggering HMB drop, dropping behavior and drop time, etc., should all meet the functional and safety requirements of HS-PSS. Additionally, structural design of HMB and its dynamics must guarantee adequate negative reactivity introduction into the core on ULOF to promptly shutdown the reactor and prevent excessive coolant heating and even boiling in the core channel. Figure 2 presents schematically the states of HS-PSS with different HMB positions.

For proper action, the following functional requirements are generally summarized:

① The absorber should be arranged at a suitable position within the HMB and possess sufficient worth. This indirectly affects the structural and dimensional design of HS-PSS and also imposes requirements on HMB motion and stroke.

② The HS-PSS is arranged equally in the core together with fuel assemblies, control rods, etc. Its design should satisfy the overall pressure drop and flow distribution requirement in the core.

③ The critical value of flowrate corresponding to the initiation of HMB dropping should be appropriately determined in HS-PSS design, so that timely dropping of the HMB when the flowrate falls below the critical value due to ULOF is ensured, and unintended HMB dropping during normal operation or other anticipated transients is prevented. So, a certain margin of coolant upward flowrate is required beyond the critical flowrate for HMB dropping. For example, 40% of nominal flowrate (40% $Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$) is designated as a target value for critical flowrate in design.

④ Once the HMB dropped down, it should be ensured not to be accidentally lifted up due to any flow fluctuations caused by malfunction or other reasons. That is, sufficient hydraulic self-tightening for the HMB seated at LWP should be ensured in HS-PSS design. For example, it is demanded that the flowrate that just lifts the HMB up from the LWP is checked for over 110% of nominal flowrate (110% $Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$) in hydraulic design of the HS-PSS.

⑤ To ensure timely introducing of negative reactivity into core through the HMB dropping, which may suppress core power, keep the average sodium temperature below a safe value, preventing sodium boiling and even severe core damage, designed HMB dropping should be smooth and rapid, and the drop time from UWP to LWP should be within a predetermined range (within 15.0 s). This is primarily achieved through appropriate structural design and optimization of HS-PSS, thereby obtaining the corresponding hydrodynamical characteristics of HMB drop.

The above items ②, ③, ④, and ⑤ are directly connected to the design features and corresponding hydro-dynamical characteristics of the HS-PSS in the present investigation, while item ① is somewhat indirectly related.

3. Hydro-Dynamic Characteristics and Behavior of HS-PSS and Implementation of the Functional Requirements

3.1. Global Dynamical Behavior and Applicable Operation Domain of HS-PSS

3.1.1. Dynamical Model of the HMB Motion in HS-PSS

As shown in Figure 2, whether HMB suspends at the UWP, drops, or sits at the LWP in the outer tube of HS-PSS depends on the coolant flowrates under normal or loss of flow accident conditions. Under various conditions, the HMB is subjected to its gravity $W$, the buoyancy $F_B$, the added mass force $F_a$ due to its accelerated movement in coolant, the hydraulic thrust $H_{Dr}$ from coolant flow, and the downward pressing or upward supporting force $F_M$ exerted by the limit structures when suspending at UWP or sitting at LWP. Additionally, during movement, HMB might come into contact with the outer sleeve, experiencing frictional force $F_m$. The motion state of HMB is the outcome of the net resultant force $F_{\mathrm{n}\mathrm{e}\mathrm{t}}$ of these external forces. Therefore, applying one-dimensional assumption and letting downward as the positive direction (as marked in Figure 2), the general HMB motion equation in the HS-PSS is obtained as

$$m_{\mathrm{H}\mathrm{M}\mathrm{B}}\ddot{y}=F_{\mathrm{n}\mathrm{e}\mathrm{t}}=W-F_B-F_a-F_{Dr}+F_M-F_m$$

(1)

on LHS of which, $m_{\mathrm{H}\mathrm{M}\mathrm{B}}$ is mass of HMB. $y$ represents displacement of HMB and $\ddot{y}$ the acceleration. On RHS, the forces exerting on HMB are:

$W$—gravitational force of the HMB (downward), which is written

$$W=m_{\mathrm{H}\mathrm{M}\mathrm{B}}g=\rho g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}},$$

(2)

where $\rho$ refers to density of HMB material and $V_{\mathrm{H}\mathrm{M}\mathrm{B}}$ volume of the HMB.

$F_B$—buoyant force on the HMB (upward), which is written

$$F_B={\rho }_{f}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}},$$

(3)

where ${\rho }_f$ refers to density of the flowing coolant. The buoyant force might be approximately written as a linear function of $y$ without excessive deviation due to weak dependence of density on temperature and the almost unchanged temperature distribution along the core channel during the period of normal operation and of HS-PSS actuation [25]. Hence

$$F_B=\left({\rho }_0+k_{T}y\right)g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}},$$

(4)

in which ${\rho }_0$ refers to the coolant density at core outlet. $k_T$ is a coefficient reflecting effect of density variation along the core region, which is dependent with the inlet–outlet temperature difference, height, and coolant thermal expansion coefficient.

$F_a$—added mass force on the moving HMB due to its relative acceleration to adjacent fluid (added mass), which is formulated as

$$F_a\left(\ddot{y}\right)=m_a\ddot{y}=C_a{\rho }_f{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\ddot{y},$$

(5)

where $m_a$ is the added mass, $C_a$ is referred to as the added mass factor. When the rod-like HMB moves in a cylindrical casing as the HS-PSS outer sleeve, the added mass coefficient $C_a$ is estimated as [20]

$$C_a={\displaystyle \frac{r_2^2+r_1^2}{r_2^2-r_1^2}}=1+2\left({\displaystyle \frac{A_{\mathrm{H}\mathrm{M}\mathrm{B}}}{A_{\mathrm{g}\mathrm{a}\mathrm{p}}}}\right),$$

(6)

where $r_1$ and $r_2$ are the outer diameter of HMB and inner diameter of the outer sleeve, $A_{\mathrm{H}\mathrm{M}\mathrm{B}}$ and $A_{\mathrm{g}\mathrm{a}\mathrm{p}}$ are cross section areas of the HMB and gap area between HMB and the outer sleeve, respectively.

$F_{Dr}$—hydraulic thrust force on the HMB, which is total resistant drags exerted by coolant on the HMB in relative motion. It consists of the pressure drop and frictional resistances $F_{Dp}$ and $F_{Df}$. With the pressure drop resistance $F_{Dp}$ being the product of a front-to-rear pressure drop across HMB ($∆p_R$) and its maximum projected area ($A_R$), $F_{Df}$ arises from the frictional shear stress ${\tau }_W$ on the wall area of HMB ($A_W$). The total hydraulic force on HMB is

$$F_{Dr}=F_{Dp}+F_{Df}=\Delta p_R{A}_{\mathrm{R}}+{\tau }_W{A}_W.$$

(7)

Defining the coefficients of pressure drop resistance and frictional resistance $C_p$ and $C_f$ as

$$C_p={\displaystyle \frac{F_{Dp}}{\frac{1}{2}{\rho }_f{v}^2{A}_R}} \mathrm{and} C_f={\displaystyle \frac{F_{Df}}{\frac{1}{2}{\rho }_f{v}^2{A}_W}},$$

(8)

where $v$ is the relative velocity between coolant and HMB ($u_f$ and $\dot{y}$) expressing as $v=u_f+\dot{y}$, the hydraulic thrust force $F_{Dr}$ is written as

$$F_{Dr}={\displaystyle \frac{1}{2}}{\rho }_{f}v\left|v\right|\left(A_R{C}_p+A_W{C}_f\right)={\displaystyle \frac{1}{2}}{C_D\rho }_{f}v\left|v\right|A_{\mathrm{e}\mathrm{v}}.$$

(9)

Here, an equivalent frontal area $A_{\mathrm{e}\mathrm{v}}=\left(A_R{C}_p+A_W{C}_f\right)/C_D$ is introduced. It is obvious that the overall flow drag coefficient $C_D$ is an implicit function of the HMB velocity $\dot{y}$ and coolant velocity $u_f$.

$F_M$—mechanical pressing or supporting force (limiting force) on HMB by either the upper or lower limit structure of the HS-PSS, depending on whether the HMB is at the UWP or LWP. At UWP or LWP, $F_M$ balances all other forces so that the HMB can stably suspend to the uppermost position or be seated at the lowermost position. It is therefore written

$$F_M\left(y\right)=\left\{\begin{array}{c}{F}_{Dr}+F_B-W, y=0\\ 0, 0<y<L\\ W-F_B-F_{Dr}, y=L\end{array}\right..$$

(10)

$F_m$—mechanical friction between HMB and outer sleeve when they make contact and are relatively moving. While the HMB dropping, it tends to be self-centered in the outer sleeve without contacting to each other [14]. So $F_m$ can well be neglected.

On the startup of a reactor, the HMB, which was already gripped by the driving mechanism and lifted to the predetermined upper position, is instructed to be released. After being released, whether in normal or abnormal operation, the HMB is passively subjected to the above-mentioned various external forces. Specifically, the forces acting on the HMB, esp. hydraulic thrust, limiting forces, etc., have a functional relationship to its hydraulic resistance characteristics and motion parameters such as displacement, velocity, and acceleration. It is with such implicit relationships of its dynamics and hydraulics that the HMB exhibits its specific motion within outer sleeve. The states include suspending or sitting at the UWP or LWP, dropping in the HS-PSS channel, which depends mainly on variation in coolant flowrate.

With such coupled dynamics as it is, a unified nonlinear formula of the one-dimensional dynamic model accounting for both dynamical and hydraulic factors for the HMB motion is thus obtained as

$$\begin{array}{c}\left(m_{\mathrm{H}\mathrm{M}\mathrm{B}}+{\rho }_0{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)\ddot{y}+{\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right){\rho }_0{A}_{\mathrm{e}\mathrm{v}}{C}_D\left({\dot{y}}^2+2u_f\dot{y}\right)+k_{T}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}y+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\ddot{y}y=m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}+\\ F_M\left(y\right)-{\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right){\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}{C}_D,\end{array}$$

(11)

in which $\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right)$ is a sign function of the relative velocity $v$.

With the dynamic model Equation (11), the various motions and possible static states of HMB in HS-PSS can be depicted. Specifically, for the required two states of HMB, one has:

① $y=0, \dot{y}=\ddot{y}=0$ (suspension state of the HMB at the UWP with or without the pressing force from the upper limit structure);

② $y=L, \dot{y}=\ddot{y}=0$ (sitting state of the HMB at the LWP with or without the supporting force from the lower limit structure).

Furtherly, allow the following dimensionless quantities:

$$Y_1={\displaystyle \frac{y}{L}}, Y_2={\displaystyle \frac{\dot{y}}{u_f}}, t^{\mathrm{*}}={\displaystyle \frac{t}{L}}{u}_f, A^{\mathrm{*}}={\displaystyle \frac{A_{\mathrm{e}\mathrm{v}}}{A_{\mathrm{r}\mathrm{e}\mathrm{f}}}}, {\rho }^{\mathrm{*}}={\displaystyle \frac{{\rho }_0}{{\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}}}.$$

Among which, $Y_1$, $Y_2$, $t^{\mathrm{*}}$, $A^{\mathrm{*}}$, and ${\rho }^{\mathrm{*}}$ refer to dimensionless displacement, velocity, time, area, and density, respectively; $A_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the projected area of HMB; and ${\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is coolant density at outlet of core region.

Then the dimensionless nonlinear dynamic equation for HMB motion is

$$M{\ddot{Y}}_1+\mathrm{s}\mathrm{g}\mathrm{n}\left(Y_2+1\right)C\left(Y_2^2+2Y_2\right)+K\left(1+K_1{\ddot{Y}}_1\right)Y_1=F-\mathrm{s}\mathrm{g}\mathrm{n}\left(Y_2+1\right)C,$$

(12)

in which the dimensionless coefficients are:

$$M={\displaystyle \frac{m_{\mathrm{H}\mathrm{M}\mathrm{B}}+{\rho }_0{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}}{L\left({\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}{A}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)}}$$

(13)

$$C={\displaystyle \frac{{\rho }_0{A}_{\mathrm{e}\mathrm{v}}}{2\left({\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}{A}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)}}{C}_D$$

(14)

$$K={\displaystyle \frac{k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gL}{u_f^2\left({\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}{A}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)}}, K_1={\displaystyle \frac{C_a{u}_f^2}{gL}}, K{K}_1={\displaystyle \frac{C_a{k}_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}}{{\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}{A}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}$$

(15)

$$F={\displaystyle \frac{m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}+F_M}{u_f^2\left({\rho }_{\mathrm{r}\mathrm{e}\mathrm{f}}{A}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)}}.$$

(16)

Apparently, Equation (12), as well as its original form, Equation (11), are non-autonomous second-order nonlinear ODEs. It is enlightening that these equations reflect dynamical characteristics of the HBM, akin to those of a nonlinear spring oscillator system. The quantities $M$, $C$, $K$, $K_1$, and $F$ are accordingly referred to as dimensionless mass, dimensionless hydraulic thrust, dimensionless stiffness arising from buoyancy with relative to the temperature distribution ($K$), dimensionless added mass effect ($K_1$), and dimensionless external force.

As a nonlinear dynamical system, HMB dynamics are governed by the model equation with the varying coefficients, and simultaneously, the coefficients still involve their inter-relations with the state variables $y$ ($Y_1$) and $\dot{y}$ ($Y_2$). For the development of HS-PSS, either the initial selection of basic parameters for the HS-PSS conceptual design or the subsequent engineering design of further compromise and optimization of structure and parameters, it is essential to investigate the global characteristics of its dynamics based on qualitative analysis of the nonlinear dynamical model [26].

3.1.2. Dynamics of the HS-PSS

(1)

Equilibrium points of HMB dynamics

Letting $Y_1$ and $Y_2$ be state variables, the state equations of the HMB dynamical system are transformed from Equation (12) as

$$\left\{\begin{array}{c}{\dot{Y}}_1=Y_2 \\ {\dot{Y}}_2={\displaystyle \frac{1}{M+K{K}_1{Y}_1}}\left[F-\mathrm{s}\mathrm{g}\mathrm{n}\left(Y_2+1\right)C{\left(Y_2+1\right)}^2-K{Y}_1\right]\end{array}\right..$$

(17)

When $K\ne 0$, the static state solution or equilibrium point ${\left(Y_1, Y_2\right)}_{\mathrm{e}\mathrm{q}}$ are unique and solved as

$$\left\{\begin{array}{c}{Y}_1={\displaystyle \frac{F-C}{K}}=Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}\\ Y_2=0 \end{array}\right..$$

(18)

Once balanced with coolant velocity $u_f$, the HMB will possibly arrive at an equilibrium point $\left(Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}, 0\right)$, where $Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}$ corresponds to an $F$, $C$, $K$-dependent suspending position, and $Y_2$ zero.

Practically for the HMB, it will suspend at UWP in HS-PSS when the upward flow velocity $u_f$ is sufficiently large during normal operation (with the absorber completely outside the core), or be seated at LWP when $u_f$ is largely reduced during ULOF accident (with the absorber completely inserted in the core). They are:

① Suspension state of the HMB at UWP ($F_M\ge 0$), with

$$Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}=0 \mathrm{or} F=C \mathrm{or} 0\le {\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u_f^{2}A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}-m_{\mathrm{H}\mathrm{M}\mathrm{B}}g=F_M.$$

(19)

② Sitting state of the HMB at the LWP ($F_M\le 0$), with

$$Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}=1 \mathrm{or} F=C+K \mathrm{or} {\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u_f^{2}A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}+k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gL-m_{\mathrm{H}\mathrm{M}\mathrm{B}}g=F_M\le 0.$$

(20)

In case the HMB locates at any position in between UWP and LWP with $F_M=0$, one has

$$0<Y_{1,Fix}<1 \mathrm{or} C<F<C+K \mathrm{or} {\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u_f^{2}A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}<m_{\mathrm{H}\mathrm{M}\mathrm{B}}g<{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u_f^{2}A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}+k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gl.$$

(21)

The equilibrium with the HMB statically suspending at any positions other than UWP and LWP is possible. In this case, the corresponding $C_D$ in Equation (21) is varied mainly for the sake of varied internal–external flow distribution of the HMB at different positions (refer to Figure 2). This adds to the complexity of the HMB suspension state. Fortunately, this suspension state is not necessarily required for both HS-PSS the design and working function.

(2)

Stability of the HMB equilibria and their dynamical behaviors

As stated, equilibrium points correspond to static suspending states of HMB. When the HMB deviates from its suspending point, whether it meets the functional design requirements of HS-PSS or not, such as whether the HMB returns or leaves the suspension point, how it returns or leaves the point, and whether the equilibrium point is stable or unstable (realistic or unrealistic), etc., all need to be understood in initial stage of HS-PSS design by analyzing stability of equilibrium points as well as dynamics around the equilibria.

For the HMB dynamical system characterized by Equation (17), when $K\ne 0$, the Jacobian matrix of the linearized state equations at equilibrium point $J_{\left(Y_{1,Fix}, 0\right)}$ and corresponding eigenvalues ${\lambda }_{1,2}$ are

$$J_{\left(Y_{1,Fix}, 0\right)}=\left[\begin{array}{cc}0& 1\\ {\displaystyle \frac{-1}{M+K_1\left(F-C\right)}}& {\displaystyle \frac{-2C}{M+K_1\left(F-C\right)}}\end{array}\right]$$

(22)

$${\lambda }_{\mathrm{1,2}}=-{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\pm {\left\{{\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2-{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}\right\}}^{1/2}.$$

(23)

Stabilities of the equilibria $\left(Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}, 0\right)$ are then determined through analyzing signs of real parts of the eigenvalues $\mathrm{R}\mathrm{e}\left({\lambda }_i\right)$. In the range of HMB displacement stroke, the various behaviors at the equilibrium point are categorized as:

① If

$$-{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}=0 \mathrm{or} C=0,$$

(24)

then two eigenvalues are imaginary, i.e., ${\lambda }_{\mathrm{1,2}}=\pm i{\left[{\displaystyle \frac{K}{M+K_{1}F}}\right]}^{1/2}$. The equilibrium point is neutral stable and called a center, which means that the point is encircled by a set of closed trajectories in the $\left(Y_1,Y_2\right)$ phase plane. They are neither drawn nor repelled by the center. Rather, they form limit cycles around it. Figure 3 illustrates an example simulation result of a postulated phase diagram satisfying the condition $C=0$ for HMB dynamics.

Physically, it corresponds to undissipated oscillation of the HMB around the suspension point, and neither converges nor diverges. However, this is neither favorable nor possible for HMB dynamics, since $C=0$ is unrealistic for real HMB movement, and it is also unacceptable for HS-PSS with the HMB oscillating periodically at any point.

② As long as with any realistic damping ($C\ne 0$), the trajectory will be either attracted or repelled by the equilibria.

If

$${\displaystyle \frac{K}{M+K_1\left(F-C\right)}}>{\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2>0 \mathrm{or} 0<C^2<MK+K{K}_1\left(F-C\right)$$

(25)

the two eigenvalues are conjugate complexes and

$${\lambda }_{\mathrm{1,2}}=-{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\pm i{\left\{{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}-{\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2\right\}}^{1/2}.$$

(26)

The equilibrium point is then a unique focus. Moreover, when $\mathrm{R}\mathrm{e}\left(\lambda \right)<0$ or $C>0$, the equilibrium point is stable (attractor), which means any displacement deviation of HMB from the equilibrium point will lead to its motion approaching back to the equilibrium; while when $\mathrm{R}\mathrm{e}\left(\lambda \right)>0$ or $C<0$, the equilibrium point is unstable (repeller), which means that once there is some slight disturbance, the HMB suspended at the point will leave away. Two example simulations of phase diagram for the two types of foci with $\mathrm{R}\mathrm{e}\left(\lambda \right)<0$ and $\mathrm{R}\mathrm{e}\left(\lambda \right)>0$ are illustrated in Figure 4. It should be mentioned that for the simulation conducted here (and also in this Section 3.1), MATLAB(R2022a)’s explicit Runge–Kutta (4,5) ode45 single-step solver is utilized to compute the dimensionless form of the HMB motion Equation (12) with certain initial conditions. Subsequently, the solutions are incorporated in the form of phase plane and phase trajectory, as demonstrated in Figure 4, which enables us to see the dependence between the solution and parameters.

For varying parameters, there exists a series of foci. If they are stable, then it is possible for the HMB to suspend at any position within the stroke in HS-PSS (with $F_M=0$). From (25), the parametric domain leading to focus satisfies

$$\left\{\begin{array}{c}\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)<k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right], \mathrm{f}\mathrm{o}\mathrm{r} \left(1+{\displaystyle \frac{1}{C_a}}\right)m_{HMB}g+F_M>{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\\ \left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)>k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right], \mathrm{f}\mathrm{o}\mathrm{r} \left(1+{\displaystyle \frac{1}{C_a}}\right)m_{HMB}g+F_M<{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\end{array}\right.$$

(27)

which approximately implies the complex and delicate relationship between the hydraulic thrust force (related to $C_D$) and inertial effect (related to $m_{\mathrm{H}\mathrm{M}\mathrm{B}}$, $C_a$).

However, with this type of focus in HBM dynamics, HMB will approach or leave the focus with dissipative oscillation, which is not acceptable for design. Therefore, focus is not considered, and the parametric domain leading to focus, defined by Equation (27), should be avoided.

③ If

$$0<{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}={\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2 \mathrm{or} 0<C^2=K\left[M+K_1\left(F-C\right)\right],$$

(28)

the two eigenvalues then degenerate to one real number, which writes

$${\lambda }_{\mathrm{1,2}}=-{\displaystyle \frac{C}{M+K_1\left(F-C\right)}},$$

(29)

then the equilibrium point is a degenerate node (a node with only one characteristic direction). With the $\lambda$ being either positive or negative, stability of the degenerate node depends on the sign of $\lambda$. The HMB dynamics phase diagram (an example with stable equilibrium) takes the form schematically as in Figure 5a.

From (28), the parametric domain leading to degenerate node satisfies

$$\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)=k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right],$$

(30)

which approximately implies the situation when the effect of inertial and that of hydraulic resistance are equal (with $F_M=0$).

④ If

$$0<{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}<{\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2 \mathrm{or} K\left[M+K_1\left(F-C\right)\right]<C^2,$$

(31)

then both the two eigenvalues are positive or negative real, which writes

$${\lambda }_{1,2}=-{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\pm {\left\{{\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2-{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}\right\}}^{1/2}.$$

(32)

The equilibrium point is either stable or unstable and they are called nodes (with two characteristic directions). The corresponding phase diagram of HMB dynamics (an example of stable node) takes the form as in Figure 5b. Parametric domain of HMB operation leading to degenerated node satisfies

$$\left\{\begin{array}{c}\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)>k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right], \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \left(1+{\displaystyle \frac{1}{C_a}}\right)m_{HMB}g+F_M>{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\\ \left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)<k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right], \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \left(1+{\displaystyle \frac{1}{C_a}}\right)m_{HMB}g+F_M<{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\end{array}\right.,$$

(33)

which also implies a rather complex and delicate relationship between the hydraulic thrust force and inertial effect, just as the situation for foci.

For the two types of equilibria of node and degenerate node, it is pointed out that one should steer clear of dynamical parameters within a large part of the above ranges for HMB design due to excessive complexity of the dynamics in vicinity of the two types of equilibrium. This is because, for example, the way the phase trajectory approaches some nodes may be not smooth and even not monotonous, and possibly there might exist a multiple-solution phenomenon around the equilibria, which is not desirable for HMB moving to or away from the suspending point.

However, dynamical behavior of the HMB in some extreme situations of node might still be acceptable. Assume

$$0<{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}\ll {\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2 \mathrm{or} K\left[M+K_1\left(F-C\right)\right]\ll C^2,$$

(34)

which corresponds the situation in which the damping factor (hydraulic thrust) ${\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2$ is much more significant compared to the stiffness factor ${\displaystyle \frac{K}{M+K_1\left(F-C\right)}}$ in the HMB dynamical system. In this case, $\mathrm{R}\mathrm{e}\left({\lambda }_1\right)~0$ and $\mathrm{R}\mathrm{e}\left({\lambda }_2\right)~-{\displaystyle \frac{2C}{M+K_1\left(F-C\right)}}$. HMB dynamics evolves to a node but with very different time scales for motions along different directions. The HMB dynamical system comes very quickly on line $Y_2=\xi Y_1$ (where $\xi ~0$), and then subsequently approaching the node is on the phase line and not throughout the phase plane. In real world, this indicates that the HMB, which is initially away from the equilibrium point, first moves towards the point with large deceleration, then approaches and stops there. Figure 6 demonstrates schematically such trajectories in a phase diagram.

This largely aligns with the characteristic requirements for the HMB approaching its suspension point and the related parametric range satisfies

$$\left\{\begin{array}{c}\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)\gg k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right], \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \left(1+{\displaystyle \frac{1}{C_a}}\right)m_{HMB}g+F_M>{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\\ \left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{A}_{\mathrm{e}\mathrm{v}}+k_T{C}_a{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)·\left({\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\right)\ll k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\left[m_{\mathrm{H}\mathrm{M}\mathrm{B}}\left(1+C_a\right)g+C_a{F}_M\right], \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \left(1+{\displaystyle \frac{1}{C_a}}\right)m_{HMB}g+F_M<{\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}\end{array}\right..$$

(35)

Therefore, in design, parameters should be judiciously selected to ensure that the hydraulic thrust (damping) factor is overwhelmingly predominant for HMB dynamics.

⑤ If ${\displaystyle \frac{K}{M+K_1\left(F-C\right)}}<0$, then ${\lambda }_{1,2}=-{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\pm {\left\{{\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2-{\displaystyle \frac{K}{M+K_1\left(F-C\right)}}\right\}}^{1/2}$, with one being negative and the other positive. The equilibrium point is a saddle. This situation is of course unacceptable for practical consideration of HS-PSS design.

For the HMB dynamical system characterized by state Equation (17), when $K=0$ (or $K$ is negligible small), which corresponds to an isothermal state in the core channel (nearly uniform coolant density along the HS-PSS), equilibrium solution is re-derived, yielding

$$\left\{\begin{array}{l}{\dot{Y}}_1=Y_2=0 \\ {\dot{Y}}_2={\displaystyle \frac{F-C}{M}}=0\end{array}\right..$$

(36)

Jacobian matrix and corresponding eigenvalues at equilibrium points are

$$J_{\left(Y_1^{*}\right)}=\left|\begin{array}{cc}0& 1\\ 0& {\displaystyle \frac{-2C}{M}}\end{array}\right|$$

(37)

$${\lambda }_{1,2}=0, {\displaystyle \frac{-2C}{M}}.$$

(38)

① If $F=C$, the HMB dynamical system possess infinitely many equilibria (they are called non-isolated fixed points), denoted as $\left(Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}^{\mathrm{*}},0\right),$ with $Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}^{\mathrm{*}}$ being an arbitrary equilibrium point within the stroke of the HMB drop ($0<Y_1<1$ and $F_M=0$), and $\left(0,0\right)$ and $\left(1,0\right)$ being the highest and lowest end equilibrium points due to the balance of pressing or supporting forces ($Y_1=0$ or $Y_1=1$, $F_M\ne 0$). In this case, Equation (36) has infinitely many equilibrium solutions, and $F=C$ actually means

$${\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u}_f^2{A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}-m_{\mathrm{H}\mathrm{M}\mathrm{B}}g=\left\{\begin{array}{c}0 , 0<Y_1<1 \\ F_M , { Y}_1=0 \mathrm{o}\mathrm{r} Y_1=1 \end{array}\right..$$

(39)

This is very natural, for it essentially implies that the resultant force of buoyancy and hydraulic thrust acting on HMB is exactly balanced by its own gravity at $Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}^{\mathrm{*}}$ within its stroke. While the HMB moves to the ends of stroke, due to adding of the limiting forces to the balance of the HMB, $Y_{1,\mathrm{F}\mathrm{i}\mathrm{x}}^{\mathrm{*}}$ keeps constants $0$ or $1$ even with further variations in $C$, $F$. Figure 7a,b present, respectively, a schematic diagram of such a kind of phase plane and an example phase diagram.

However, since $\mathrm{R}\mathrm{e}\left({\lambda }_i\right)$ is either zero or positive, the infinitely many equilibria will be either unstable or neutral stable within the HMB stroke, which is unfit for them to be stably suspended. However, for the upper and lower limit positions, the situation is different. Stability of these two equilibria will change due to the addition of the upper pressing force or the lower supporting force.

An example simulating movement of a HMB with different initial positions and velocities (denoted with [$Y_1, Y_2$]) is illustrated in Figure 8. It is seen that the HMB with a different initial state will move to suspend at different positions ($Y_1=Y_1^{*}$, $Y_2=0$). Even in the event that $F=C$ holds, the final HMB suspending position $Y_1^{*}$ varies with initial state, which is unacceptable. See Figure 8b.

Therefore, dynamical behavior of HMB for the $F=C$ situation is unacceptable, let alone, practically, the drag coefficient $C_D$ can never be fixed when the HMB is dropping all along its stroke, and $F=C$ is hence almost unrealistic to hold for all points within the stroke of HMB.

② If $F\ne C$, which is more realistic for HMB, there is no equilibrium point throughout the stroke, until an upper or lower limiting position is reached and the HMB is stopped.

Figure 9 gives sample simulations of the HMB motion phase diagram for fixed parametric conditions (without considering constrains from the upper and lower limits). It is found in Figure 9a that, for $F>C$, all phase trajectories flow towards $+\infty$ position at a velocity of $Y_{2,a}>0$. It is understood that the HMB will drop when $F>C$. If the supporting force at the lower limit position is considered, it corresponds to the HMB dynamic behavior of dropping to the lower limit and being seated there, and from Figure 9b, one can see similarly that when $F<C$, the HMB will move upwards with the velocity of $Y_{2,a}<0$ and eventually suspend at the upper limit position.

Summarizing the above analysis, distribution of various parametric domains of the HMB dynamics as well as corresponding equilibrium point types are shown in Figure 10. As analyzed, the possible parametric domain that is applicable for the HMB functional design is located in the shaded area (including the adjacent vertical axis), whose parameters are necessarily defined by

$$0\le {\displaystyle \frac{K}{M+K_1\left(F-C\right)}}\ll {\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2 \mathrm{or} 0\le K\left[M+K_1\left(F-C\right)\right]\ll C^2.$$

(40)

A special applicable case is when

$$K=0,$$

(41)

which corresponds to zero temperature rise in the core region (isothermal channel) situation, and it is obvious that this almost joins the domain defined by (40), with similar dynamic behavior in reality.

It is interesting to find that, in this combined area, the hydraulic thrust effect takes overwhelming advantages over the buoyancy with core temperature rise and added mass effect, and the hydraulic thrust effect is the dominant controlling factor of HMB movement, which right aligns with the primary function of HS-PSS.

Some parameters for the initial design of the prototypic HS-PSS (including HMB) for this study are given as following:

-

Outer diameter of HMB: $d_{\mathrm{H}\mathrm{M}\mathrm{B}}~0.09 \mathrm{m}$; length of HMB: $L_{\mathrm{H}\mathrm{M}\mathrm{B}}~1.15 \mathrm{m}$; volume of HMB: $V_{\mathrm{H}\mathrm{M}\mathrm{B}}~0.007 {\mathrm{m}}^3$; inner diameters of the outer sleeve: $d_{\mathrm{O}\mathrm{S}-\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}~0.094 \mathrm{m}$, $d_{\mathrm{O}\mathrm{S}-\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}~0.092 \mathrm{m}$; equivalent frontal area of HMB: $A_{\mathrm{e}\mathrm{V}}~0.0064 {\mathrm{m}}^2$; maximum displacement of HMB dropping: $L~0.9 \mathrm{m}$; and mass of HMB: $m_{\mathrm{H}\mathrm{M}\mathrm{B}}~22 \mathrm{k}\mathrm{g}$.

-

HS-PSS inlet coolant temperature: $350 °\mathrm{C}$; HS-PSS exit coolant temperature: $550 °\mathrm{C}$; inlet flow velocity $u_f=2.6 \mathrm{m}/\mathrm{s}$.

-

Other parameters estimated under the operation conditions (typical value or orders of magnitude): $C_D=6.87$; $C_a=5.28$; $k_T=0~52.82; \mathrm{a}\mathrm{n}\mathrm{d} {\rho }_f=820.37~867.91 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$.

With the above parameters, it is calculated that: $M=10.817$; $C=3.435$; $K=0.043$; $K_1=4.043$; and $F=4.160$. Therefore, we have

$${\displaystyle \frac{K}{M+K_1\left(F-C\right)}}=0.0032\ll {\left[{\displaystyle \frac{C}{M+K_1\left(F-C\right)}}\right]}^2=0.062,$$

which obviously satisfies the requirement defined by (40).

(3)

Important dynamic performances of HS-PSS and design implications

According to functional requirements defined in Section 2, three important functional aspects related to dynamic performances of the HS-PSS are considered for conceptual design. They are connected with HMB’s suspension state at UWP, sitting state at LWP, and HMB dropping movement.

① UWP suspension state

For suspension state of the HMB at UWP, (19) gives

$$F_M={\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u_f^{2}A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}-m_{\mathrm{H}\mathrm{M}\mathrm{B}}g\ge 0.$$

(42)

Hence, when the HMB remains in a suspension state at UWP, the following should be met

$$u_f\left(t\right)\ge \sqrt{{\displaystyle \frac{2\left(m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)}{{\rho }_0{A}_{\mathrm{e}\mathrm{v}}{C}_D}}}=u_{f,\mathrm{c}\mathrm{r}\mathrm{i}},$$

(43)

where the flow velocity $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ that just suspends the HMB is called critical velocity for suspension and the corresponding flowrate $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$ ($u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}=Q_{\mathrm{c}\mathrm{r}\mathrm{i}}/\left({\rho }_0{A}_{\mathrm{e}\mathrm{v}}\right)$ is called critical flowrate. When $u_f\left(t\right)$ exceeds $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$, the HMB remains suspended; if $u_f\left(t\right)$ is less than $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$, the HMB starts dropping.

Suppose that the HMB is suspending at UWP ($y=0$) and coolant flow through the HS-PSS is experiencing a coast-down, which is characterized with continuous flow velocity $u_f\left(t\right)$ decreasing just as what happens during ULOF accident. If $u_f\left(t\right)$ coasts down to that lower than the critical velocity $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ corresponding to the suspending position, the force balance of the HMB is broken and it starts dropping. When ULOF occurs, it is necessary that $u_f\left(t\right)$ and $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ intersects. During normal operation, $u_f\left(0\right)$ should be a certain amount higher than $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$; whereas, once loss of flow observed, $u_f\left(t\right)$ should drop to below $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ for HMB dropping, as seen in Figure 11a. There are still other scenarios, that is, the flowrate is either consistently above or always below the critical flowrate (for the latter situation, HMB cannot be kept at UWP solely by hydraulic thrust and requires grasping by the driving mechanism). If $u_f\left(t\right)$ is higher than $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$, the HMB will keep suspending at UWP, while with $u_f\left(t\right)$ smaller than $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$, the HMB will drop whenever it is released by the driving mechanism. Figure 11b,c schematically demonstrates these situations.

Under actual accident condition, flow velocity $u_f$ is not fixed. In particular, during ULOF, $u_f\left(t\right)$ decreases according to the main pump coast-down curve. It should also be noted that, the value of $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ mainly depends on the designed path configuration within the HS-PSS, which specifically leads to the variation in $C_D$ with the HMB moving, and according to (43), the greater $C_D$ is, the smaller $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ might be.

In ULOF accident, if critical flowrate $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$) is too low, HMB dropping is delayed, preventing timely introduction of negative reactivity into the core, which hampers its prompt response against ULOF. At the same time, to prevent unintended HMB drop due to flow fluctuation during normal operation, $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$ must not be too high. Therefore, the critical flowrate $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$ should be appropriately compromisingly determined by design, usually set at a proper percentage $D\%$ of the nominal flow $Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$ ($u_{f,\mathrm{n}\mathrm{o}\mathrm{m}}=Q_{\mathrm{n}\mathrm{o}\mathrm{m}}/\left({\rho }_0{A}_{\mathrm{e}\mathrm{v}}\right)$) in the channel, ie., $D\%·Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$. At this time, the drag coefficient corresponding to the critical flowrate is

$$C_{D, \mathrm{c}\mathrm{r}\mathrm{i}}\ge {\displaystyle \frac{2\left(m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}\right)}{{\rho }_0{A}_{\mathrm{e}\mathrm{v}}{\left(D\%·u_{f,\mathrm{n}\mathrm{o}\mathrm{m}}\right)}^2}}.$$

(44)

With other parameters ($m_{\mathrm{H}\mathrm{M}\mathrm{B}}$, ${\rho }_0$, $V_{\mathrm{H}\mathrm{M}\mathrm{B}}$, and $A_{\mathrm{e}\mathrm{v}}$) of a HMB fixed and the coast-down flow curve (shown in Figure 12a,b), Figure 12b presents the simulated critical velocities under $Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$ and $40\%Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$ ($D\%=40\%$), the corresponding $C_{D, \mathrm{c}\mathrm{r}\mathrm{i}}$s, as well as the intersection $A$ of $u_f=u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ line and $u_f=u_f\left(t\right)$ curve, which implies a dropping point. It is seen that the higher $C_{D, \mathrm{c}\mathrm{r}\mathrm{i}}$ and the smaller $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ are, the later intersection and later HMB dropping are observed.

In HS-PSS design, it is essential to ensure that geometrical design meets the HMB dropping condition by (44) through iterative simulation on hydraulics within the HS-PSS.

② Sitting state at LWP

When the HMB is seated in the buffer section, coolant will flow upward through the HMB from the outside of the buffer section without forming a frontal impinging on the HMB, which will greatly reduce the drag coefficient $C_D$ and the resultant hydraulic thrust. For sitting state of the HMB at LWP, on the other hand, (20) gives

$$F_M={\displaystyle \frac{1}{2}}{C}_D{\rho }_0{u_f^{2}A}_{\mathrm{e}\mathrm{v}}+{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}+k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gL-m_{\mathrm{H}\mathrm{M}\mathrm{B}}g\le 0,$$

(45)

from which the lowest coolant upward flow velocity that lifts the HMB up from LWP is

$$u_f\left(t\right)\le \sqrt{{\displaystyle \frac{2\left(m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}-k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gL\right)}{{\rho }_0{A}_{\mathrm{e}\mathrm{v}}{C}_D}}}=u_{f, \mathrm{c}\mathrm{r}\mathrm{i},\mathrm{S}\mathrm{T}},$$

(46)

in which the coolant velocity $u_{f,\mathrm{c}\mathrm{r}\mathrm{i},\mathrm{S}\mathrm{T}}$ is another critical velocity that just lifts the HMB up.

Therefore, in order to meet the functional requirement ④, sufficient flow margin should be reserved for hydraulic self-tightening in design consideration. If it is required that the minimum of the coolant velocity $u_{f,\mathrm{c}\mathrm{r}\mathrm{i},\mathrm{S}\mathrm{T}}$ that otherwise might re-lift the HMB from where it is seated be $S\%·u_{f,\mathrm{n}\mathrm{o}\mathrm{m}}$ (corresponding to $S\%·Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$) as targeted by design criteria, the maximum equivalent drag coefficient $C_D$ by now is derived from (46) as

$$C_D\le {\displaystyle \frac{2\left(m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}-k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gL\right)}{{\rho }_0{A}_{\mathrm{e}\mathrm{v}}{u}_{f, \mathrm{c}\mathrm{r}\mathrm{i},\mathrm{S}\mathrm{T}}^2}}={\displaystyle \frac{2\left(m_{\mathrm{H}\mathrm{M}\mathrm{B}}g-{\rho }_{0}g{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}-k_T{V}_{\mathrm{H}\mathrm{M}\mathrm{B}}gL\right)}{{\rho }_0{A}_{\mathrm{e}\mathrm{v}}{\left(S\%·u_{f, \mathrm{n}\mathrm{o}\mathrm{m}}\right)}^2}}.$$

(47)

In fact, satisfaction of (47) by $C_D$ ensuring hydraulic self-tightening performance of HMB at LWP is implemented through design adjustments on the specific geometric structure (mainly the flow paths jointly formed by HS-PSS outer sleeve, the HMB front structure and the buffer cup) and hence related hydraulic characteristics. Test validation is conducted for every modification.

③ HMB drop and drop time

As seen in Figure 2, during the HMB dropping period, the ratio of narrow-gap to wide-gap length between the outer sleeve and HMB increases with the HMB displacement increasing, which leads to a decreasing trend of $C_D$ during most of its dropping process and hence, decreasing hydraulic thrust. This makes HMB drop faster and faster until the head of the HMB enters the buffer section (buffer cup) at the late dropping stage. When the HMB enters, it squeezes coolant in the buffer cup upwards to form a reaction force, which slows down dropping.

It is one of the functional requirements for the HS-PSS to ensure HMB drop time within a certain range, which is connected to dynamic behavior of HMB dropping. HMB dropping dynamics are affected by gravitational, buoyancy, hydraulic thrust, and added mass forces that vary with its movement and affect each other.

For the initial design of HS-PSS, the following group of parameters are taken: $m_{\mathrm{H}\mathrm{M}\mathrm{B}}=22 \mathrm{k}\mathrm{g}$; $d_{\mathrm{H}\mathrm{M}\mathrm{B},\mathrm{o}\mathrm{u}\mathrm{t}}=0.09 \mathrm{m}$; $L_{\mathrm{H}\mathrm{M}\mathrm{B}}=1.15 \mathrm{m}$; $V_{\mathrm{H}\mathrm{M}\mathrm{B}}=0.007 {\mathrm{m}}^3$; $L=0.9 \mathrm{m}$; $d_{\mathrm{O}\mathrm{S}-\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}=0.094 \mathrm{m}$; $C_a=5.28$; and $C_D=6.87$ in the range estimated by empirical relation for the annular gap; $A_{\mathrm{e}\mathrm{v}}=0.0064 {\mathrm{m}}^2$; $T_{\mathrm{i}\mathrm{n}}=350 °\mathrm{C}$; $T_{\mathrm{o}\mathrm{u}\mathrm{t}}=550 °\mathrm{C}$; $u_f=2.6 \mathrm{m}/\mathrm{s}$; and ${\rho }_f\left(y\right)=\left[820.37+\left(47.54y\right)/L\right] \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$.

From the balance condition (19) for HMB suspension, suspension mass for the referred parameters is $m_{\mathrm{H}\mathrm{M}\mathrm{B},\mathrm{c}\mathrm{r}\mathrm{i}}=\left({\displaystyle \frac{C_D{A}_{\mathrm{e}\mathrm{v}}}{2g}}{u_f}^2+V_{\mathrm{H}\mathrm{M}\mathrm{B}}\right){\rho }_0=20.0 \mathrm{k}\mathrm{g}$. If $m_{\mathrm{H}\mathrm{M}\mathrm{B}}\le m_{\mathrm{H}\mathrm{M}\mathrm{B},\mathrm{c}\mathrm{r}\mathrm{i}}$, the HMB keeps suspending; while if $m_{\mathrm{H}\mathrm{M}\mathrm{B}}>m_{\mathrm{H}\mathrm{M}\mathrm{B},\mathrm{c}\mathrm{r}\mathrm{i}}$, the HMB will drop.

Sensitivity analysis through simulation is conducted. Except that $m_{\mathrm{H}\mathrm{M}\mathrm{B}}$ takes a series of values within $20~23 \mathrm{k}\mathrm{g}$ and ${\rho }_f={\rho }_0=820.37 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ (corresponding to $T_{\mathrm{i}\mathrm{n}}=T_{\mathrm{o}\mathrm{u}\mathrm{t}}=550 °\mathrm{C}$), all other parameters take the above referred values for simulation. Figure 13 presents results of displacement, velocity, and drop time for various HMB masses.

It is seen that HMB mass has obvious influence on its dropping motion. In Figure 13a, the HMB with a mass of $20 \mathrm{k}\mathrm{g}$ cannot drop. As it increases from $20.5 \mathrm{k}\mathrm{g}$ to $23.0 \mathrm{k}\mathrm{g}$, drop time reduces rapidly from $13.88 \mathrm{s}$ to $3.95 \mathrm{s}$. Correspondingly, it is seen from Figure 13b,c that the greater the HMB mass, the more rapidly its drops. It is also observed that extremely rapid decrease in drop time occurs if HMB mass is near the suspension mass value $m_{\mathrm{H}\mathrm{M}\mathrm{B},\mathrm{c}\mathrm{r}\mathrm{i}}$, while the decreasing trend of drop time slows down with further mass increasing.

Once HMB dropping starts, it will become deeper and deeper inserted into the core, and coolant density variation, due to the temperature rise, has some impact on the buoyancy of the HMB dropping. Figure 14 presents a comparison of simulation results by solving the HMB state equations, concerning the HMB dropping with different masses ($m_{\mathrm{H}\mathrm{M}\mathrm{B}}=20~23 \mathrm{k}\mathrm{g}$) and different temperature rises in the core region ($T_{\mathrm{i}\mathrm{n}}-T_{\mathrm{o}\mathrm{u}\mathrm{t}}=350 °\mathrm{C}-350 °\mathrm{C}$ and $350 °\mathrm{C}-550 °\mathrm{C}$). So, with $m_{\mathrm{H}\mathrm{M}\mathrm{B}}=23 \mathrm{k}\mathrm{g}$ and $T_{\mathrm{i}\mathrm{n}}=350 °\mathrm{C}$, and $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ of $350 °\mathrm{C}$ increasing to $550 °\mathrm{C}$, the HMB drop time decreases from $5.50 \mathrm{s}$ to $4.19 \mathrm{s}$. Therefore, the lower the temperature level of coolant and thus the greater the buoyancy on HMB, the shorter the drop time that is observed.

By solving HMB state equations, drop motion with a different combination of $C_D$ and $u_f$ is simulated, with $T_{\mathrm{i}\mathrm{n}}=T_{\mathrm{o}\mathrm{u}\mathrm{t}}=350 °\mathrm{C}$ and all other parameters set to the referred values. It is easy to have: if $C_D=2$ fixed, then the corresponding critical velocity $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}=4.95 \mathrm{m}/\mathrm{s}$; while if $u_f=2.5 \mathrm{m}/\mathrm{s}$ is fixed, the critical drag coefficient $C_{D,\mathrm{c}\mathrm{r}\mathrm{i}}=7.85$. It is evident that the HMB will definitely drop as long as the actual $u_f$ and $C_D$ are smaller than the two critical values simultaneously.

Figure 15 shows the influences of $C_D$ and $u_f$ combinations on $F_{Dr}$ during dropping. Additionally, the calculated $C_D$ and $u_f$ sensitivity on HMB dropping are presented in Figure 16.

It is obvious from Figure 15 that the higher the drag coefficient $C_D$ and the higher the coolant flow velocity $u_f$, the greater the hydraulic thrust force $F_{Dr}$ is. Moreover, $F_{Dr}$ does not increase linearly with the increase in $C_D$ and $u_f$ due to the simultaneous impact of HMB dropping velocity $\dot{y}$.

Additionally, it is observed from Figure 16 that the smaller $C_D$ and $u_f$ are, and the further away they are from their critical values $C_{D,\mathrm{c}\mathrm{r}\mathrm{i}}$ $\left(7.85\right)$ and $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$ $(4.95 \mathrm{m}/\mathrm{s})$, the greater the HMB dropping velocity and acceleration that are observed, resulting in a shorter drop time. Conversely, with higher $C_D$ and $u_f$, and the nearer they are to the critical values, the dropping velocity $\dot{y}$ and acceleration $\ddot{y}$ are reduced, and the drop time is extended. In particular, as $C_D$ and $u_f$ are approaching their critical values, both the dropping velocity and acceleration decline sharply, and drop time increases significantly. Regarding the variation in drop time with $C_D$ and $u_f$, some results are summarized in

Table 1. Drop time of HMB with different $C_D$ and $u_f$.

${\mathit{C}}_{\mathit{D}}=2$			${\mathit{u}}_{\mathit{f}}=2.5 \mathbf{m}/\mathbf{s}$
$\frac{{\mathit{u}}_{\mathit{f}}}{{\mathit{u}}_{\mathit{f},\mathbf{c}\mathbf{r}\mathbf{i}}} (\mathbf{\%})$	${\mathit{u}}_{\mathit{f}} (\mathbf{m}/\mathbf{s})$	$\mathbf{Drop} \mathbf{Time} (\mathbf{s})$	$\frac{{\mathit{C}}_{\mathit{D}}}{{\mathit{C}}_{\mathit{D},\mathbf{c}\mathbf{r}\mathbf{i}}}, (\mathbf{\%})$	${\mathit{C}}_{\mathit{D}}$	$\mathbf{Drop} \mathbf{Time} (\mathbf{s})$
0%	0	1.522	0%	0	1.514
10%	0.495	1.542	10%	0.785	1.626
30%	1.485	1.636	30%	2.355	1.930
50%	2.475	1.842	50%	3.925	2.434
70%	3.465	2.311	70%	5.495	3.505
90%	4.455	4.166	90%	7.065	8.362
98%	4.851	11.993	98%	7.693	37.570

.

It is seen that both $C_D$ and $u_f$ are below their critical values during HMB dropping. The closer they are to the critical values, the longer the drop time; influencing range and sensitivity of $C_D$ and $u_f$ on dropping are somewhat different.

In summary, it is required that automatic variation in $C_D$ during HMB drop should be achieved through HS-PSS hydraulic structure design according to variation characteristics of $u_f$ (expected coast-down flow curve). This is essential to ensure that the HMB maintains its required suspension, meets the specified drop time and drop velocity requirement, and on the other hand, maintains a sufficiently low $C_D$ at the lowermost region to fulfill the hydraulic self-tightening function.

3.2. Dynamical Behavior of HS-PSS Coupling with Its Hydraulics

Static and dynamic characteristics of HS-PSS are determined by the temporal and spatial variations of the forces acting upon the HMB. Achieving control of the forces, especially the hydraulic thrust, on HMB with the changing in flowrate and displacement is crucial for the HS-PSS to initiate appropriate and timely responses during ULOF accidents, and thereby fulfills requirements for critical suspension flowrate, drop time, and hydraulic self-tightening. Therefore, it is essential, through proper design of the special and complex HS-PSS hydraulic structures, to manipulate internal–external flow distribution and pressure drop on HMB with its movement and thus the time–space variation of $C_D$, so as to control the hydraulic thrust.

3.2.1. Hydraulic Structure of HS-PSS

For a certain reactor with its normal operation and loss of flow conditions, core temperature rise and coolant velocity are basically determined. Moreover, the shape and size, as well as material of the HS-PSS components (primarily the outer sleeve and HMB) are overall constrained by form and size of the assembly installation channel and of the absorber bundle within HMB. There is quite limited room and flexibility for sensible adjustments on HMB mass ($m_{\mathrm{H}\mathrm{M}\mathrm{B}}$), volume ($V_{\mathrm{H}\mathrm{M}\mathrm{B}}$), frontal area ($A_{\mathrm{e}\mathrm{v}}$), and other related parameters such as $k_T$, ${\rho }_0$, etc. Therefore, in HS-PSS design, it is essential to focus on the hydraulic structure adjustment to achieve appropriate control on the drag coefficient $C_D$ or hydraulic thrust force $F_{Dr}$ corresponding to coolant flowrate variation in the core. On the one hand, this must satisfy the requirements for the HMB upper suspension state, drop time, and dropping velocity. On the other hand, it has to ensure that $C_D$ at LWP is sufficiently low to fulfill the requirements of the hydraulic self-tightening function. The expected trend of $C_D$ or $F_{Dr}$ with HMB displacement is schematically shown in Figure 17.

As schematically shown in Figure 18, the primary hydraulic parts related to the function of HS-PSS are composed of a stationary outer sleeve and a mobile HMB. The shape and size of the HS-PSS outer sleeve are similar to those of fuel assembly. As a key hydraulic component for safety function in HS-PSS, the HMB is shaped as a hollow rod structure. The HMB wall is a supporting structure to separate internal and external flow and install the absorber bundle so that it is carried up and down in HS-PSS. The absorber bundle is fixed to the HMB wall through upper and lower grid plates. An orifice plate structure is set above the bundle to adjust the resistance of the internal flow path of HMB. There are openings at the bottom and top of the HMB as coolant inlets and outlets, and connections of the HMB internal and external flow paths. The HS-PSS outer sleeve is also provided with holes at its lower and top ends for coolant entering and exiting.

By setting such structures as holes, sleeves, orifice, and rod bundles in the assembly, flow and pressure drop among the paths can be distributed, thereby influencing hydraulic thrust and HMB drop. In Figure 18, the following hydraulic structures play regulating roles:

-

From the HMB side, by reasonably setting size and number of coolant inlet and outlet holes, adjusting the orifice holes inside HMB and thus changing flow resistance of HMB internal flow paths, proper distribution of internal and external flow as well as pressure drop is achieved.

-

For the outer sleeve, by setting its inner diameter which changes along axial length, the length ratio of narrow to wide annular paths between the HMB and outer sleeve decreases with HMB drop displacement, and hence, continuous reduction in flow resistance outside the HMB is obtained.

-

At the very late stage of dropping, the process of HMB entering the buffer cup also makes it end-buffered.

Thus, a dynamic complex coolant flow path is established between the outer sleeve and HMB, “auto” adapting the HMB‘s suspension, dropping, and sitting.

In addition, with the coolant inlet of the assembly at the lower end of the outer sleeve, the overall pressure drop and flow across HS-PSS can be adjusted through modifications to its inlet hole size and internal structures.

3.2.2. Hydraulic Modeling and HS-PSS Hydro-Dynamic Behavior Calculation

The static and dynamic behavior of HMB for its safety function realization is closely coupled with its hydraulic characteristics. The influence of hydraulic thrust $F_{Dr}$, or simply of drag coefficient $C_D,$ on HMB motion is the key. Through hydraulic structural design of the HMB, $F_{Dr}$ or $C_D$ interacts in a complex way with HMB displacement and coolant flow variation. Therefore, only with hydraulic modelling and analyzing $F_{Dr}$ or $C_D$ as well as related influencing factors can dynamic behavior of HMB be well understood and its design effectively conducted and optimized.

Figure 19 schematically demonstrates a designed HS-PSS at different states with related fluid network structures. As seen, a complex fluid network is formed with the diverse interconnective structures and combinations within and between the HS-PSS outer sleeve and HMB. Specific flow distribution among the inter-connected small paths is established within the network and the motion of HMB keeps resulting in alterations to the flow path structures and their connections.

Set reference frame on the HMB with the downward direction as positive. A one-dimensional fluid network model of HS-PSS is established for hydraulic thrust calculation. The network of HS-PSS is, as demonstrated in Figure 19, decomposed into connected/interconnected small path elements, with each element composed of nodes and channels.

Let the flow path number between any two nodes $m$-$n$ in the network be $i$, then one has the discrete form of pressure drop equation for the flow path $i$, which writes

$$P_m-P_n=\mathsf{\Delta }{p}_{\left(\lambda +\xi \right),i}-{\rho }_f{\displaystyle \frac{\mathit{\partial }{v}_i}{\mathit{\partial }t}}{l}_i-{\displaystyle \frac{v_m^2-v_n^2}{2}}+∆p_{l,i},$$

(48)

in which, $\mathsf{\Delta }{p}_{\left(\lambda +\xi \right),i}$ is the friction plus form pressure loss of path $i$; ${\rho }_f{\displaystyle \frac{\mathit{\partial }{v}_i}{\mathit{\partial }t}}{l}_i$ the pressure drop caused by flow velocity variation over time in path $i$, written in time–space discrete form as ${\rho }_f{\displaystyle \frac{\mathit{\partial }{v}_i}{\mathit{\partial }t}}{l}_i={\rho }_{f,i}{l}_i\left[v_i^{\left(k\right)}-v_i^{\left(k-1\right)}\right]/∆t$; $\left(v_m^2-v_n^2\right)/2$ is the pressure drop due to spatial acceleration and $∆p_{l,i}={\rho }_{f,i}g{l}_i$ the gravitational pressure drop. Here, the flow velocity $v$ for pressure drop estimation is the relative velocity, namely, $v_i=u_i+\dot{y}$. Then, irreversible pressure drop term is

$$\mathsf{\Delta }{p}_{\left(\lambda +\xi \right),i}=\mathsf{\Delta }{p}_{\lambda ,i}+\mathsf{\Delta }{p}_{\xi ,i}={\displaystyle \frac{{\rho }_f}{2}}\left({\displaystyle \frac{f_i{l}_i}{D_i}}+k_n\right){\displaystyle \frac{{Q_i}^2}{{A_i}^2}},$$

(49)

in which $f_i$ is the friction coefficient of path $i$; $k_n$ the local resistance coefficient at node $n$; $l_i$, $D_i$, and $A_i$ the length, hydraulic diameter, flow area of path $i$; and $Q_i$ refers to relative volumetric flowrate in path $i$.

In particular, take the HMB dropping state as an example, lengths of the end-to-end-connected narrow and wide annular channels between HMB and the outer sleeve vary with HMB displacement. In this case, structure of the connecting paths spanning nodes 10-3-4-11 changes with the HMB displacement $y$, resulting in variations in frictional pressure drops $\mathsf{\Delta }{p}_{\lambda ,5}$ and $\mathsf{\Delta }{p}_{\lambda ,7}$ as

$$∆p_{\lambda ,5}={\displaystyle \frac{{\rho }_f}{2}}{\displaystyle \frac{f_5}{D_5}}{\displaystyle \frac{{Q_4}^2}{{A_5}^2}}\left(L_5-y\left(t\right)\right)={\displaystyle \frac{{Q_4}^2}{2{\rho }_f}}{\displaystyle \frac{f_5}{D_5{A_5}^2}}\left(L_5-y\left(t\right)\right)$$

(50)

$$∆p_{\lambda ,7}={\displaystyle \frac{{\rho }_f}{2}}{\displaystyle \frac{f_7}{D_7}}{\displaystyle \frac{{Q_4}^2}{{A_7}^2}}\left(L_7+y\left(t\right)\right)={\displaystyle \frac{{Q_4}^2}{2{\rho }_f}}{\displaystyle \frac{f_7}{D_7{A_7}^2}}\left(L_7+y\left(t\right)\right).$$

(51)

(1)

Friction coefficient relationships

According to typical design of the HS-PSS, the main types of flow path are classified as: pipes (circular/non-circular), bundles, and annuli (round-round/hexagonal-round; with movable/immovable inner surfaces). Friction coefficient $f_{0,i}$ for these types of flow paths are:

-

Pipes: For circular pipes, the correlations recommended by Idelchik [27] are used to calculate $f_{0,i}$ under different flow conditions. For noncircular tubes, the correlations used are exactly the same, except that diameter of circular pipe is replaced by hydraulic diameter.

-

Bundles: the Cheng and Todreas correlation [28] is used for calculation friction pressure drop in the absorber bundle flow paths.

-

Annuli: for annular gap with immovable inner surface [29],

$$f_{0,i}={\displaystyle \frac{64}{\mathrm{R}\mathrm{e}}}·{\displaystyle \frac{{\left(1-d_1/d_2\right)}^2}{1+{\left(d_1/d_2\right)}^2-\left[1-{\left(d_1/d_2\right)}^2\right]/\ln \left(d_1/d_2\right)}}, \mathrm{R}\mathrm{e}\le 2000$$

(52a)

$$f_{0,i}=\left({\displaystyle \frac{0.02d_1}{d_2+0.98}}\right)\left[{\displaystyle \frac{1}{f_{0,i-\mathrm{c}\mathrm{i}\mathrm{r}}}}-0.27\left({\displaystyle \frac{d_1}{d_2}}\right)+0.01\right], \mathrm{R}\mathrm{e}>2000,$$

(52b)

in which $d_1$ and $d_2$ are the inner and outer diameter of annuli, respectively; $f_{0,i-\mathrm{c}\mathrm{i}\mathrm{r}}$ is the friction coefficient of a circular pipe with $d_2$ in diameter at the same $\mathrm{R}\mathrm{e}$.

For an annular gap with movable inner surface, $f_{0,i}$ in the laminar regime ($\mathrm{R}\mathrm{e}\le 2000$) is

$$f_{0,i}={\displaystyle \frac{64}{\mathrm{R}\mathrm{e}}}·{\displaystyle \frac{{\left(1-d_1/d_2\right)}^2}{1+{\left(d_1/d_2\right)}^2-\left[1-{\left(d_1/d_2\right)}^2\right]/\ln \left(d_1/d_2\right)}}\left[1+{\displaystyle \frac{\dot{y}}{u}}\left({\displaystyle \frac{1}{1-{\left(d_1/d_2\right)}^2}}-{\displaystyle \frac{1}{2\ln \left(d_1/d_2\right)}}\right)\right],$$

(53a)

where $\dot{y}$ is velocity of the moving surface, and $u$ is fluid velocity within the annuli which is $u=4Q_i/\left[\pi \left(d_2^2-d_1^2\right)\right]$.

In a turbulent regime ($\mathrm{R}\mathrm{e}>2000$), Formula (52b) is boldly used which, was then posteriorly validated as acceptable for the narrow gap geometry of HS-PSS [30]. Thus,

$$f_{0,i}==\left({\displaystyle \frac{0.02d_1}{d_2+0.98}}\right)\left[{\displaystyle \frac{1}{f_{0,i-\mathrm{c}\mathrm{i}\mathrm{r}}}}-0.27\left({\displaystyle \frac{d_1}{d_2}}\right)+0.01\right].$$

(53b)

Considering that the characteristic dimensions of the flow path elements are relatively small in HS-PSS, and considering the potential impact of non-ideal factors such as un-fully developed flow, the authors carried out a series of hydraulic tests on frictional resistance of the components within typical structural dimension and flowrate ranges. By measuring pressure drop across the tested components, correction factor $C^{*}$ was obtained and the actual friction coefficients are determined [13]

$$f_i=C^{*}{f}_{0,i}.$$

(54)

(2)

Local resistance coefficient relationships

The node element structure in the HS-PSS network primarily includes sudden contraction, sudden expansion, converging tee branch, diverging tee branch, right-angle elbow, small hole, throttling and grid plate, etc. Among the node elements, local resistance coefficients $k_n$ of sudden contraction/expansion, converging/diverging tee branch, and right-angle elbow are calculated through the relationships recommended by Idelchik [27], while $k_n$ for some special components such as special orifice and grid plate have to be decided by tests [30]. Moreover, in the HS-PSS flow path structure, some components are combined, among which these combinations are often encountered, such as lower-end plug small hole, throttling element outlet hole, grid plate–rod bundle, grid plate-throttling outlet hole, and lower-end plug grid plate–rod bundle throttling. Considering interaction effects among tightly adjacent components, these component combinations are also tested for combined resistance, and related correction $K^{*}$ factors were obtained.

Local pressure drop for the components or combinations are calculated as [13,30]

$$\mathsf{\Delta }{p}_{\xi ,i}=K^{*}{\sum }_{j=1}^n{k}_{j,i}{\displaystyle \frac{{\rho }_f{u}_{j,i}^2}{2}}, j=1, 2,\cdots ,n.$$

(55)

(3)

Flow supplemented by squeezing effect during HMB dropping

As shown in Figure 20, HMB will produce a squeezing effect when it drops due to the limited space, leading to the fluid below being pressed into the upper channel, resulting in an extra flowrate $∆Q\left(t\right)$ for gap flow correction as

$$∆Q\left(t\right)={\rho }_f\dot{y}{A}_R.$$

(56)

When the HMB is suspending at UWP or seated at LWP, $\dot{y}=0$, and thus $∆Q\left(t\right)=0$.

(4)

Addressing the buffer effect

The cross section of HS-PSS buffer structure is schematically shown in Figure 21.

In late stage of HMB dropping when it reaches the buffer section, the inlet flow directly bypasses the HMB lower-end plug and flows upward. Once the HMB enters the buffer cup, structure of the fluid network changes. At the same time, when the HMB falls into the cup and forms a relative displacement $y$ with it, the fluid inside is squeezed first to the annular gap and then into the upper flow channel.

If the dropping speed of HMB is $\dot{y}$, then the volumetric flowrate $Q_b$ into the gap is

$$Q_b=A_R\dot{y}.$$

(57)

The narrow gap flow between the buffer cup and HMB will result in substantial pressure loss, which contributes to a pronounced pressure difference cross the HMB and adds to the hydraulic thrust $F_{Dr}$. This leads to buffering effect, which slows down the HMB dropping. Applying a momentum theorem to coolant within the control volume shown in Figure 21, the hydraulic thrust for this buffer section $F_{Dr, i}$ is calculated as

$$F_{Dr,i}={\displaystyle \frac{{\rho }_f{A}_R^2{\dot{y}}^2}{A_0-A_R}}.$$

(58)

(5)

Fluid network model of HS-PSS hydraulics and integrated hydraulic calculation

An overall fluid network of HS-PSS in the HMB dropping stage is shown in Figure 22. Based on Kirchhoff law, control equations addressing the flow and pressure drop distribution among the integrated flow paths and nodes are formulated as

$$\left\{\begin{array}{l}∆P-{∆P}_1=0 \\ {∆P}_{a1}-{∆P}_{a2}=0 \\ {∆P}_{b1}-{∆P}_{b1}=0 \\ Q_1+∆Q-Q_2-Q_3=0 \\ Q_2-Q_4-Q_5=0 \\ Q_3+Q_5-Q_6=0 \\ Q_4+Q_6-∆Q-Q_7=0 \end{array}\right. \mathrm{or} \left\{\begin{array}{l}∆P-\sum {∆P}_{1,2~6}=0 \\ \sum {∆P}_{9,12~13}-\sum {∆P}_{9,10~13}=0 \\ \sum {∆P}_{10,3~11}-\sum {∆P}_{10,13~20,11}=0 \\ Q_1+∆Q-Q_2-Q_3=0 \\ Q_2-Q_4-Q_5=0 \\ Q_3+Q_5-Q_6=0 \\ Q_4+Q_6-∆Q-Q_7=0 \end{array}\right..$$

(59)

Based on the pressure drop relationships given above, the first three pressure drop equations are formulated about $Q_i$. By employing Newton iteration method to solve the transcendental equations in terms of $Q_i$, flow and pressure drop distribution across each flow path are obtained. Then, the HS-PSS total pressure loss $∆P$ is calculated as

$$∆P=P_6-P_1=\sum {∆p}_{\left(\lambda +\xi \right), i}+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{{\rho }_{f1}{Q}_1}{{\rho }_{f6}{A}_6}}\right)}^2-{\left({\displaystyle \frac{Q_1}{A_1}}\right)}^2\right]+\sum {\rho }_{f,i}g{l}_i,$$

(60)

and thus is the lumped hydraulic thrust $F_{Dr}$ or $C_D$ on the HMB obtained.

(6)

Coupled hydro-dynamics of HS-PSS: code development and testing validation

HS-PSS behavior and its hydro-dynamic characteristics are calculated and validated with the relevant computer code and test facility as follows:

① PSSD code for HS-PSS hydro-dynamic analysis

A one-dimensional analysis and design code for hydraulically suspended passive shutdown subassembly entitled PSSD is developed [15] and improved, which aims to be applied not only to greatly improve the computation efficiency of the HS-PSS hydro-dynamical analysis, but also to facilitate the optimized selection of design schemes and ensure the accuracy and reliability of HS-PSS design calculations.

The code incorporates the dynamic model for HMB motion and the HS-PSS hydraulic model. Its primary computational process involves solving the dynamic Equation (11) for the HMB with quasi-steadily mapping of HS-PSS hydraulic model solutions. In solving the nonlinear dynamic equations, time is discretized. For each time step $∆t$, hydraulic characteristics of the HS-PSS and thus the forces on the HMB are calculated (by Newton-Raphson iterative method) and transferred as load to the dynamic equation solution module, and then motion of the HMB is analyzed (by the efficient dynamic analysis algorithm entitled Newmark-β method [31]). Through step-by-step solution technique, static and dynamic responses of the HMB, such as suspension at UWP, dropping, and sitting at LWP, are obtained for evaluation. Since the entire process of HMB dropping is within 15 s as required, and the drop speed is of the order of 0.2 m/s or so, it is both reasonable and feasible to select a suitable $∆t$ (usually $∆t~0.001 \mathrm{s}$ in practice) and employ the above-mentioned coupling method for calculating the HMB motion process.

PSSD code is utilized in the assembly design phase for analyzing the hydraulic characteristics of the HS-PSS, as well as the kinetic and dynamic performance of HMB in the postulated ULOF accident.

② VERCH facility for test validation

In order to validate the predicting and analyzing capabilities of PSSD code, as well as to confirm the hydro-dynamic performance of the designed prototypic test sample of the HS-PSS, a water loop test facility entitled VERCH was constructed [14].

VERCH is a closed hydraulic loop for a hydraulic test with a storage water tank (with heating), water pumps, vertical test section, plate heat exchanger, surge tank, and related pipes and valves, which is schematically shown in Figure 23. During testing, deionized water is heated in the water storage tank, then flows through the pump, vertical test section, the heat exchanger for cooling, and finally returns to the tank it started out in. The HS-PSS prototypic test sample is mounted vertically in the test section. Testing flowrate through the test section is controlled either by the combination of valves, variable frequency pumps, and the bypass for a fixed flowrate, or by the variable frequency pump simulating a predetermined coast-down flow. At the same time, system pressure is controlled by the surge tank and water temperature in the loop is regulated through the storage tank heater, the preheater, and the plate heat exchanger.

The test section simulates the HS-PSS path for coolant flow in the reactor. It includes the HS-PSS prototype, the simulated bottom grid plate, the gripping device at the top, and the outer sleeve for accommodation of the parts, as shown in Figure 24. A series of dynamic signals are picked up for measurement during the test, which include the inlet/outlet temperature, mass flowrate through the test section, pressure/differential pressure along the test section, forces on the HS-PSS, and HMB under static conditions. Synchronously, displacement of the HMB, as well as its movement, are obtained through the glass window with high-speed camera videoing and image processing frame by frame.

A gripping mechanism is attached to the test section from the top, using a pneumatic gripper and gear linkage mechanism. During the tests, the gripper can grasp the HMB from the head, making it lift, stay, and lower and release it underwater. These actions facilitate the suspension, dropping, and hydraulic characteristics tests of the HMB. For detailed operation and procedure for these specific tests, refer to [14]. With the tests, such indictors and curves such as critical flow, dropping curve (HMB displacement vs. time curve), hydraulic characteristics curve ($∆P$–$Q$ curve), and hydraulic self-tightening performance for the HMB can be experimentally acquired. Moreover, it should be mentioned that for the HMB dropping tests, it includes the dropping test under fixed flow and the under coast-down flow.

In the facility, geometric ratio of the test section and the channel to their prototypes is 1:1. Additionally, by regulating water temperature and inlet flowrate, hydrodynamic similarity between the testing and prototypic condition is ensured in terms of equivalent hydraulic resistance, i.e., it keeps $\mathrm{R}\mathrm{e}$ numbers equal.

The physical properties of coolant involved in the hydraulic thrust acting on HMB mainly include dynamic viscosity and density. In practical reactors, the inlet sodium temperature of HS-PSS is $360 °\mathrm{C}$, and the average axial sodium temperature inside is $420 °\mathrm{C}$. Hence, considering the principle that the kinetic viscosity is closest to the prototypic liquid sodium, in the VERCH water loop test, water at $84 °\mathrm{C}$ to $93 °\mathrm{C}$ can be selected for testing to simulate sodium at $360 °\mathrm{C}$ to $420 °\mathrm{C}$. That is, hydraulic thrust ${\left(F_{Dr}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$ tested on the VERCH facility with water at temperature of $84 °\mathrm{C}$/$93 °\mathrm{C}$ equals ${\left(F_{Dr}\right)}_{\mathrm{N}\mathrm{a}}$ with sodium at a temperature of $360 °\mathrm{C}$/$420 °\mathrm{C}$. Furtherly, with similarity, VERCH tested critical flowrate ${\left(Q_{\mathrm{c}\mathrm{r}\mathrm{i}}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$ or ${\left(Q_{m,\mathrm{c}\mathrm{r}\mathrm{i}}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$, pressure difference ${\left(∆p\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$, and drop time $t_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$ with water at $84 °\mathrm{C}$/$93 °\mathrm{C}$ can be converted to its counterpart quantities with sodium at $360 °\mathrm{C}$/$420 °\mathrm{C}$ by [30]

$${\left(Q_{\mathrm{c}\mathrm{r}\mathrm{i}}\right)}_{\mathrm{N}\mathrm{a}}={\left(Q_{\mathrm{c}\mathrm{r}\mathrm{i}}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}\sqrt{{\displaystyle \frac{{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}{{\rho }_{\mathrm{N}\mathrm{a}}}}}$$

(61a)

$${\left(Q_{m,\mathrm{c}\mathrm{r}\mathrm{i}}\right)}_{\mathrm{N}\mathrm{a}}={\left(Q_{m,\mathrm{c}\mathrm{r}\mathrm{i}}\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}\sqrt{{\displaystyle \frac{{\rho }_{\mathrm{N}\mathrm{a}}}{{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}}}$$

(61b)

$${\left(∆P\right)}_{\mathrm{N}\mathrm{a}}={\displaystyle \frac{{\rho }_{\mathrm{N}\mathrm{a}}}{{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}}{\left(∆P\right)}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}$$

(61c)

$$t_{\mathrm{N}\mathrm{a}}=t_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}\sqrt{{\displaystyle \frac{{\rho }_{\mathrm{H}\mathrm{M}\mathrm{B}}-{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}{{\rho }_{\mathrm{H}\mathrm{M}\mathrm{B}}-{\rho }_{\mathrm{N}\mathrm{a}}}}}.$$

(61d)

Therefore, with the VERCH water loop facility, tests on HS-PSS hydraulic characteristics, on HMB suspension and critical flowrate, on hydraulic self-tightening performance, and tests of HMB dropping under both fixed and coast-down flowrates are carried out. Hydraulic characteristics of HS-PSS and dynamic behaviors of HMB are studied in the tests, which simultaneously involves direct differential pressure, temperature, and flow measurements, HMB displacement and dropping velocity measurements through visualized image processing, and measurements of the pressing and support forces with HMB at UWP and LWP.

3.2.3. Hydraulic Characteristics of the HS-PSS

In the test for $∆P-Q_m$ hydraulic characteristics with the HMB fixed at the UWP, variation in total pressure drop across the prototypic HS-PSS test sample with the $93 °\mathrm{C}$ water mass flowrate increasing stepwise in the range of $0~4.3 \mathrm{k}\mathrm{g}/\mathrm{s}$ is measured. Meanwhile, results calculated for exactly the same conditions are obtained. Figure 25 demonstrates the comparison between test data and calculation. It is seen that for the HMB fixed at a certain position in the HS-PSS channel, pressure drop $∆P$ across the HMB increases with the inlet mass flowrate $Q_m$ increasing. The higher the $Q_m$, the faster the $∆P$ increases. Calculation results are generally in good agreement with test data, which primarily verifies the hydraulic modelling and related code. The acquired steady hydraulic characteristics may be applied in evaluating the HMB’s UWP suspension performances, as well as related design optimization.

Pressure drop $∆P$ (thus drag coefficient $C_D$ or hydraulic thrust $F_{Dr}$) plays a crucial role in suspension and drop motion of HMB. It is obviously determined by not only the total flowrate $Q$, but also the flow and pressure drop distribution within the HS-PSS. Conceptually, a special design is made for the HS-PSS outer sleeve so that the gap size between the sleeve and HMB, and thus the related $C_D$ or $F_{Dr}$, varies with HMB displacement, through which the functional requirements for suspension, drop, and hydraulic self-tightening of HMB are properly met. Figure 26 presents the calculated $∆P-Q_m$ curves for the HS-PSS testing sample with HMB fixed at different positions (water temp.: $93 °\mathrm{C}$).

It is seen from Figure 26 that, for the same flowrate, the larger the HMB displacement, the smaller the pressure drop. Therefore, the flowrate that can just make the HMB critically suspended at UWP, namely the critical flowrate $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$, cannot at all keep the suspension of HMB at lower positions. Additionally, as inferred from Figure 26, when the HMB is located at LWP (HMB displacement being 900 mm), the flowrate that can just lift up the HMB (marked as $Q_{\mathrm{c}\mathrm{r}\mathrm{i},\mathrm{S}\mathrm{T}}$ in the figure) is much higher than $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$. This indicates that the design of the annular gap between HMB and the outer sleeve in the present HS-PSS sample, which is narrow at the top and wide at the bottom, achieved the goal of enabling the HMB with hydraulic self-tightening performance at LWP.

Moreover, among the contributors to the overall pressure drop across the HS-PSS, the small path elements with high proportions of influence on pressure drop are identified through sensitivity calculation to be the small gap (path 4–11 in Figure 22) and internal throttling hole (path 18–19 in Figure 22). By adjusting the size of path 4–11 (gap) and of path 18–19 (hole), influence of the key component dimension variations on the hydraulic characteristics of the HS-PSS (with HMB at UWP or LWP) is demonstrated in Figure 27.

It is evident that when HMB is suspended at UWP, variations in both sizes have some influence on the overall pressure drop, with the gap size effect being much more noticeable; while with the HMB seated at LWP (with zero length of gap path, and quite small flowrate into HMB in this situation), changing the throttling hole size makes almost no difference in pressure drop. This implies that modifying the dimensions of the gap and hole both may somewhat affect the critical flowrate for HMB dropping, yet do almost no help to its self-tightening performance. Therefore, during the optimization design, the most sensitive and effective means to alter critical flowrate is to adjust the gap (4–11) size in fulfillment of the functional requirement ③, whereas adjusting the size of the internal throttling hole (18–19) comes to the second and might serve as a fine adjustment. However, these two adjustments have almost no significant effect on changing the hydraulic self-tightening characteristics of HMB at LWP for fulfilling the functional requirement ④.

3.2.4. Hydro-Dynamic Behaviors of the HS-PSS in Action

In addition to HS-PSS’s hydraulic characteristics, which are related to variation in flow and pressure drop under various conditions, dynamic response to ULOF accident is also examined in design. Three main indicators of HMB actions examined for implementation of related functional requirements include: critical flowrate, HMB drop time, and self-tightening flowrate at LWP.

(1)

UWP suspending state: critical flowrate

The resultant force of hydraulic thrust, gravity, and buoyancy of HMB in suspension state is zero. The coolant velocity/flowrate corresponding to this state is the critical velocity $u_{f,\mathrm{c}\mathrm{r}\mathrm{i}}$/critical flowrate $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$. When coolant flow through the suspending HMB is lower than the critical value, HMB start to drop, while for higher flow, HMB keeps suspending.

By finely reducing flowrate stepwise through the prototypic HS-PSS test sample with HMB suspending at UWP, an (averaged) $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$ is determined by repeated tests (water temp.: $84 °\mathrm{C}$, $93 °\mathrm{C}$). Comparison between the tested and calculated critical flowrates for the same condition is given in

Table 2. Comparison between the tested and calculated critical flowrates.

No.	Displacement (mm)	Temperature (℃)	$\mathbf{Tested} {\mathit{\rho }}_{\mathit{f}}{\mathit{Q}}_{\mathbf{c}\mathbf{r}\mathbf{i}}$ (kg/s)	$\mathbf{Calculated} {\mathit{\rho }}_{\mathit{f}}{\mathit{Q}}_{\mathbf{c}\mathbf{r}\mathbf{i}}$ (kg/s)	Error (%)
1	0	84.0	1.79 ± 0.02	1.82	1.7
2	70	84.0	1.90 ± 0.02	1.93	1.6
3	0	93.0	1.82 ± 0.02	1.85	1.6
4	70	93.0	1.95 ± 0.02	1.98	1.5

. The calculated $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$ seems slightly higher than test data, and the overall error is within 2%, which is acceptable. The tested and calculated $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}$ are also used as an indicator for assessing test sample’s fulfillment of prescribed functional requirement in respect to critical flowrate in HS-PSS design (present design target: $Q_{\mathrm{c}\mathrm{r}\mathrm{i}}\approx 40\% Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$).

(2)

LWP sitting state: self-tightening performance

Figure 28 presents the tested and calculated $∆P-Q_m$ curves for the prototypic HS-PSS testing sample when HMB is at LWP and UWP (water temp.: $84 °\mathrm{C}$). The corresponding $F_{Dr}-Q_m$ curves calculated by PSSD code are given in Figure 29.

It is seen in Figure 28 that the $∆P-Q_m$ curves with the HMB at UWP and LWP are quite different. It is also easy to infer that the overall drag coefficient decreases with dropping till it reaches the minimum when HMB is at LWP (since $∆P$ decreases with displacement increasing when dropping, as demonstrated in Figure 26). In Figure 29, the intersection for the $F_{Dr}$ curve with HMB at UWP with the horizontal “gravity minus buoyance line” corresponds to the critical mass flowrate $Q_{m,\mathrm{c}\mathrm{r}}$, while that for the $F_{Dr}$ curve with HMB at UWP corresponds to the self-tightening flowrate $Q_{m,\mathrm{c}\mathrm{r},\mathrm{S}\mathrm{T}}$ (out of the figure), which is much higher than $Q_{m, \mathrm{c}\mathrm{r}}$. This means that a much higher flowrate and thus greater thrust force is required to lift up the HMB from LWP, which implies a good self-tightening performance. For the present design, its fulfillment of prescribed self-tightening requirement with margin is hence confirmed (present design target: $Q_{\mathrm{c}\mathrm{r}\mathrm{i},\mathrm{S}\mathrm{T}}>110\% Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$).

Force condition on the HMB at LWP is meanwhile measured and observed for validation. Online calibration for the waterproof force sensor is pre-test conducted based on force balance of the HMB under zero-flowrate condition. During test, the measured hydraulic thrust of HMB is directly obtained. The variation in measured hydraulic thrust $F_{Dr}$ with the stepwise increasing of flowrate is shown in Figure 30. As can be seen, at a flowrate of 5.0 kg/s, which is well over $110\% Q_{\mathrm{n}\mathrm{o}\mathrm{m}}$, the average hydraulic thrust on HMB is 26.4 ± 0.1 N, which is significantly less than the weight of the test sample. Therefore, the hydraulic self-tightening performance of the prototypic HS-PSS which satisfies the corresponding functional requirement is also empirically demonstrated.

(3)

HMB dropping and drop time

Drop time of HMB is an important performance affecting realization of HS-PSS safety function. Dynamic behavior of HMB dropping is directly affected by the $F_{Dr}$ or $C_D$ variation it experiences, which is in turn closely related to the hydraulic structure and dimensions of the HS-PSS.

Aiming at believable, measurable observation into the actual dynamic process of HMB dropping in response to the ULOF accident, tests with regulating prescribed coast-down flowrate for the prototypic HS-PSS testing sample were carried out, utilizing the VERCH hydraulic test loop under conditions closely resembling practical applications.

Figure 31 illustrates a representative set of HMB drop test data with the coast-down water flow of $84 °\mathrm{C}$.

The HMB first remains suspending at UWP. With flowrate coasting down and the subsequent change in hydraulic characteristics by HMB displacement, overall pressure drop across the HS-PSS gradually decreases. Once flowrate reaches the critical value of 1.80 kg/s, the HMB starts to drop. The dropping process can be, as depicted in Figure 31, roughly divided into four phases, namely, Phases A, B, C, and D: Within the initial period of around 2 s, the HMB experiences a sudden force state change from suspension, thus enters a transitional motion period. A positive jump of HMB acceleration is observed, companying an overshoot with velocity rapidly increasing from zero to about $0.03 \mathrm{m}/\mathrm{s}$ (Phase A). Then, possibly due to reaction to the abrupt HMB acceleration from surrounding fluid, a swift recovery of hydraulic thrust is subsequently triggered, which stabilizes the HMB motion. From then on, HMB enters a relatively stable falling phase, characterized by fairly stable acceleration and gradual increasing velocity (Phase B). As the HMB further drops, its relative velocity with the fluid intensifies, accompanied by changes in flow path structure within the HS-PSS testing sample. It might give rise to a bit faster decrease in pressure drop and further increase in drop velocity, possibly resulting in onset of drag crisis (now with ${\mathrm{R}\mathrm{e}}_{\left(\dot{y}-u_f\right)}$ estimated $~{10}^5$) and a slightly violent fluctuation of acceleration. Consequently, the HMB enters the accelerated dropping stage (Phase C). As dropping continues, the lower end of HMB enters buffer section. The flow path outside the HMB is basically the wide-gap section, resulting in a decrease in overall drag coefficient. The combined effect of flow coasting down and drag coefficient decreasing leads to a significant decrease in total pressure drop across the HS-PSS. By now, the HMB enters the buffering period until it finally reaches LWP (Phase D), and the final drop velocity comes to about 0.4 m/s. The drop time is well within 15.0 s (the present drop time target).

For the purpose of validating the PSSD capability of prediction, the drop process is simulated by the code with almost the same coast-down flowrate during HMB dropping as in the test. Comparison between code simulation and test data is shown in Figure 32, which well demonstrates the capability of the code simulating HMB dropping. As seen, there exists a small discrepancy between the simulation and test data, with slight variations in their critical flow values. Consequently, start points of the two drop curves differ a little and drop time difference between the results is about 0.22 s. Allowing for the many uncertain factors in the code settings and test measurements, the minor difference can well be acceptable for analysis and design.

In order to investigate dynamics of HMB dropping under typical ULOF coast-down flow condition, a set of PSSD simulation results of the dropping process is given in Figure 33. The temporal variations of HMB displacement, related forces on the HMB, as well as the prescribed coast-down flowrate are demonstrated. Among the given calculated forces, the gravity $W$, buoyancy $F_B$, hydraulic thrust $F_{Dr}$, added mass force $F_a$, and the upward resultant force $F_{up}$ expressed as $F_{up}=F_{Dr}+F_B+F_a-W$ are encompassed.

It is seen that as flowrate at the entrance of HS-PSS decreases continuously with the HMB at UWP, the hydraulic thrust $F_{Dr}$ exerted on HMB correspondingly decreases. When the upward resultant force $F_{up}$ gradually decreases to zero, the HMB starts dropping, and the flowrate right at this moment corresponds to the critical flowrate for HMB suspension at UWP.

As the HMB drops, its increasing drop velocity results in an elevated relative velocity $v$ to surrounding fluid, which thereby tends to enhance the hydraulic thrust $F_{Dr}$. Nevertheless, it seems that the reduction trend of $F_{Dr}$ due to the changing of flow paths and the descending of coast-down flow outweighs the limited contribution of increased $v$. Consequently, $F_{Dr}$ continues to decrease with the HMB’s falling, yet the decreasing trend becomes significantly less steep. Although the added mass force increases from zero as dropping starts, its influence on the dropping motion seems negligible. Additionally, buoyancy $F_B$ is much smaller compared to gravity and hydraulic thrust, and is expected to have a relatively minor influence on dropping motion.

Furthermore, Figure 34 compares the HMB dropping curves across two different temperature conditions. As shown, there is a certain difference in HMB drop curve between the two temperature tests, but the difference is not significant. At $84 °\mathrm{C}$, the HMB starts dropping when flowrate reaches about 1.80 kg/s, and the drop time is 10.65 s; at $93 °\mathrm{C}$, the HMB begins to drop when the flowrate is about 1.82 kg/s, and the dropping takes 10.48 s. It needs to be stated that, the drop time of the HMB for water at $84 °\mathrm{C}$ is 10.65 s, which is extended to a time of 10.56 s for sodium at $360 °\mathrm{C}$, and the functional requirements for drop time are met.

To search for information about geometrical effects on HMB dropping dynamics for design optimization, Figure 35 presents the sensitive results of size effect of some key flow structures in the HS-PSS (as structured in Figure 22) on HMB dropping. The key structures picked out for study include, as indicated in Figure 22, narrow annuli 4–11, small paths 17–18, 19–20, throttle 18–19, and inlet hole 9–12. For each flow path structure in the HS-PSS, with the related size of the present design as reference value, five sizes are selected: the reference value, $\left(1\pm 10\%\right),$ and $\left(1\pm 20\%\right)$ of the reference value. For the same coast-down flowrate, HMB dropping processes are simulated with PSSD code, and sensitivities of the structural dimension effects on starting point of HMB dropping (corresponding to the critical flowrate) as well as the drop time are examined.

It is seen that alternations in certain key structural dimensions might significantly affect both critical flowrate and drop time, such as narrow annuli 4–11 and the internal throttle hole 18–19. As for the small paths 17–18 and 19–20, which are upstream and downstream the throttle hole, and the inlet hole 9–12, although they contribute to the overall pressure drop of the HS-PSS to a certain extent, their size variations exert a relatively minor influence on the dropping process. The effects of sensitivities of structural size on HMB dropping behavior (critical flowrate and drop time) are summarized in Figure 36.

4. Conclusions

Based on its working principle, behaviors and functions of HS-PSS with HMB suspension or dropping under normal operation or ULOF accident conditions of CFR 600 are analyzed in this article. The functional requirements for conceptual design and corresponding indicators such as critical flowrate, drop time, hydraulic self-tightening performance, and overall hydraulic characteristic curve of HS-PSS are summarized.

With the nonlinear dynamical model of HMB established, qualitative analysis on the equilibrium solution properties of the HMB motion equation in the HS-PSS is conducted. This enable us to derive the global dynamics of HS-PSS and arrive at its applicable parameter domain for design, which is tailored to the functional requirements. Within the parameter domain applicable for structural design, hydraulic thrust is found to be the dominant controlling factor for HMB movement, which is the primary working principle of HS-PSS. It preliminarily provides a practical guidance for the parameter selection in HS-PSS conceptual design.

Furtherly, in detail, integrated theoretical dynamical and hydraulic modeling, code simulation and dedicated test validation on the hydraulic characteristics and hydro-dynamic behaviors, as well as working performance of a prototypic HS-PSS design are carried out. Fulfillment of the functional requirements in terms of the relevant performance indicators, namely that of critical flowrate, drop time, hydraulic self-tightening performance, and the overall HS-PSS hydraulic characteristics, is confirmed. Additionally, sensitivity analysis on the effects of the key flow-path structures of HS-PSS is conducted and specific crucial structural sizes sensible to the hydro-dynamic performance are identified. This provides assistance in HS-PSS-detailed design and structure finalization. Understandings in this dynamical and hydraulic aspects of HS-PSS are quite helpful for optimizing its design in response to ULOF.