Molten Salt Pump Journal-Bearings Dynamic Characteristics Under Hydrodynamic Lubrication Conditions

Liu, Yuqi; Chen, Minghui; Che, Shuai; Burak, Adam

doi:10.1115/1.4064336

Abstract

A reliable high-temperature molten salt pump is critical for the development of Fluoride-salt-cooled, High-temperature Reactors (FHRs). By supporting the rotating journal, the appropriate journal bearing can ensure that the high-temperature molten salt pump runs smoothly and efficiently in the high-temperature fluoride salt over a long period of time. However, many bearing candidates served well for only a short period and experienced several issues. Moreover, the alignment of the molten salt pump journal bearings is a key factor for the molten salt pump's long-term steady running. In the long-term operation, a misalignment in the journal bearing can result in vibrations and excessive wear on the bearing surface of the molten salt pump. The journal bearing dynamic characteristics can be used to accurately assess the journal misalignment. Therefore, it is necessary to investigate the detailed journal-bearing dynamic behavior under the high-temperature hydrodynamic fluoride salt lubrication conditions for FHR applications. In this study, a small amplitude vibration is superimposed on a steady-running journal bearing to simulate the molten salt operating conditions. A Fortran 90 code has been written for the journal-bearing dynamic behavior analysis. The code was verified using the numerical data reported in the literature. The code is then employed to predict the dynamic coefficients of high-temperature fluoride salt hydrodynamic lubricated journal-bearing with various Sommerfeld numbers. These journal-bearing dynamic coefficients can be used to provide guidelines in the design of molten salt pumps.

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1 Introduction

Bettis and Briant first proposed the molten salt reactor (MSR) for aircraft propulsion in the 1940s [1]. To pursue this idea, the Oak Ridge National Laboratory (ORNL) in the U.S. initiated extensive research culminating in the Aircraft Reactor Experiment in 1954 [1]. In the second half of the 1950s, molten salt technology was incorporated into the U.S. civil nuclear code, and ORNL initiated the molten salt reactor experiment, which demonstrated the viability of molten salt reactors [2]. Following molten salt reactor experiment, ORNL designed a prototype molten salt breeder reactor in the early 1970s, which included continuous fuel reprocessing to reduce neutron losses captured in fission products [2]. The MSR has a number of potential advantages over traditional nuclear reactor designs. The salt used in the MSR has a high melting and boiling point. Therefore, MSRs must be operated at higher temperatures than conventional water-cooled reactors, which increases thermal efficiency. The fuel utilization rate is improved, and the waste generation rate is reduced. The risk of a serious accident is reduced because the MSR can operate at low pressure, and the salt remains liquid [2]. Two of the most promising MSR system design concepts have been developed [3]. First, the molten fast-spectrum reactor Molten Fast-Spectrum Reactor (MSFR) is a liquid-fuel fast neutron reactor, where fissile and possibly fertile materials are dissolved in the molten salt coolant [3]. Second, fluoride-salt-cooled high-temperature reactor (FHR) utilizes solid fuel, often in tri-structural isotropic fuel particles embedded in graphite or other matrices [4]. FHR and MSFR are part of the Generation IV suite of advanced reactor designs, which aim to improve safety, sustainability, and economics compared to previous generations [5]. However, MSFR and FHR have distinct features, operations, and fuel cycles. FHR is a nuclear reactor that combines solid fuel and a coolant of fluoride salts. Primary and intermediate circuit temperatures, crucial for designing IHXs, still need to be defined [6]. The supercritical steam cycle demonstrates the optimum performance [7]. The expected fuel salt temperature range at the core inlet is 649–700 °C, and the fuel salt temperature range at the core outlet is 750–800 °C [7,8]. However, these systems' novel features introduce unique challenges, especially regarding materials, operational temperatures, and the reactor's dynamic response to various conditions. Addressing these challenges requires in-depth research, innovation, and robust engineering solutions to realize the full potential of these advanced reactor designs. However, the development of MSRs also faces several technical challenges, including the need for materials that can withstand the highly corrosive, high-temperature salt environment [5]. The molten salt pump journal bearing faces significant challenges when operating for a long term in a high-temperature, corrosive salt environment. With prolonged operation in the molten salt environment, the journal bearing becomes susceptible to seizure, wear, instability, and catastrophic accidents. This challenge will be one of the major obstacles in the development of MSRs, from small experimental reactors to commercial reactors. FLiBe, a mixture of lithium fluoride (LiF) and beryllium fluoride (BeF₂) has been selected as the coolant for MSRs. The melting and boiling points of the FLiBe salt are 458.85 °C and 1430.40 °C, respectively [9]. The FLiBe salt has a good neutron absorption cross section and a low corrosion rate for metallic alloys, which helps prolong the reactor's life [10]. It also has good breeding properties that allow it to produce additional fuel when irradiated with neutrons [10].

Long-shaft vertical pumps are standard practice in MSRs [8]. The long-shaft vertical pump discharge and suction connections are close to the nuclear reactor core [11]. Figure 1 shows that the long shaft can separate the driving motor from the high temperature and radio activity. The long shaft will provide enough space to accommodate radiation shielding. The journal bearings are mechanical components designed to support the post radially. The salt pump shaft is immersed in journal bearing and lubricated with FLiBe salt [9]. The rotation of the shaft induces pressure variation, and then, the lubricant is sucked into the gap between the post and the bearing surfaces [11]. In this case, a vertical centrifugal sump-type salt pump was used in the FLiBe salt. The hydrodynamic journal bearing is a sliding bearing. The part of the shaft supported by the path is called the journal, and the role that matches the journal is called the bushing, which supports the mechanical components, as shown in Fig. 1. The journal bearing is applied in heavy-load conditions and is accessible for fabrication [9]. The long-shaft vertical pump is the most promising candidate for FHRs.

Fig. 1

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Schematic of a long-shaft vertical pump

The lubricant separates the journal and the bushing surfaces without direct contact, which can significantly reduce friction loss and surface wear. The radial clearance of the bearing determines how these forces behave, making it a crucial parameter in rotordynamic control. The clearance between the shaft and bushing is from 80 to 200 μm [9]. The lubricant will be filled in the clearance. The lubricant film also has a specific ability to absorb vibration. The film forces that are generated due to the lubrication mechanism have a significant impact on the stiffness and damping of the system. However, the misalignment will spur reaction forces and excitation of natural machine frequencies. As defined by Muszynska, misalignment refers to the inconsistency between the shaft's axis of rotation and the bearing's geometric axis. This inconsistency can be minute or pronounced, but even small degrees of misalignment can have significant effects [12]. Misalignment can disrupt the usual pressure profile of the lubricant film, leading to a decline in the bearing's overall performance [13]. The misalignment can also lead to bearing wear due to uneven load, vibration, reduced bearing life, and thermal overheating. Misalignment can accelerate wear, particularly at bearing edges. The load concentrates on smaller areas, increasing the bearing surfaces' stress. Misalignment also can induce vibrations that can manifest as noise. Over time, this can lead to fatigue failures [14,15]. Misalignment causes uneven heat distribution in the bearing, creating localized hot spots. This can be particularly harmful in systems like molten salt pumps already operating in high-temperature, corrosive salt environments. Hydrodynamic film forces show that the film forces, and consequently the stiffness and damping, are susceptible to misalignment. Thus, two parameters that need to be defined, the pressure applied to the displacement and the resistance to motion, are the most critical factors in controlling rotordynamics [12].

The stiffness coefficient is expressed as the ratio of the force applied to the resulting displacement. The damping coefficient represents the resistance to motion and is associated with the damping force that opposes the system's velocity. The stiffness and damping coefficients determine the dynamic coefficients of journal bearings. The bearing design has evolved to achieve higher damping and stiffness. Due to the journal motion within the bearing clearance, these properties correspond to changes in the hydrodynamic force generated in the lubricant film. The instability can increase the motion without limit and lead to destructive consequences for the machine. If the instability is uncorrected immediately, it will quickly cause destructive vibration and catastrophic failure. A rotor's integrity and reliability depend significantly on its bearings' dynamic characteristics [14]. Adjusting the journal bearings' stiffness and damping coefficients can avoid instability problems, raise or fine-tune a system's natural frequencies outside the operating speed range, and increase the instability threshold beyond the operating speed range [15]. The stiffness of the journal-bearing system is mainly determined by the bearing support stiffness acting in series with shaft stiffness. The bearing damping properties usually determine the system's damping almost entirely [13]. Such journal bearings are used for their adequate load support, good damping characteristics, and absence of wear if properly designed and operated. Therefore, the stiffness and damping coefficients of the journal bearings play a critical role in influencing the long-term steady-state performance of the primary pump in fluoride salt-cooled high-temperature reactors, especially when operating with high-temperature fluoride salt lubricants. Despite the extensive general bearing literature, there is a noticeable lack of studies addressing misalignment in molten salt pumps, revealing a distinct research gap. The unique operating conditions of FHR pumps, characterized by elevated temperatures and the corrosive nature of the fluid, underscore the increased impact of misalignment and necessitate focused research in this area. Finally, accurately predicting journal bearings' stiffness and damping coefficients play a decisive role in long-term steady operation with high-temperature fluoride salt lubricants in FHRs primary pumps.

The detailed insights of the Fortran90 code highlight the multidimensional challenges and considerations in the design and maintenance of molten salt pump journal bearings. Understanding and optimizing the dynamic coefficients can ensure the longevity, reliability, and efficiency of the pump, saving high costs and preventing catastrophic failures. The Fortran90 code was developed to evaluate the dynamic coefficients, the first time it has been used for molten salt pumps. In this paper, the dynamic characteristics (stiffness and damping coefficients) under the high-temperature fluoride salt environment are obtained by using the Fortran90 code. It can accurately predict the dynamic characteristics, and numerical data obtained by Lund validate the code. This research provides a rich overview of the importance of journal-bearing design and the influence of hydrodynamic film forces on rotor dynamics, especially in the context of FHR pumps. In addition, the stiffness and damping coefficients have been tabulated and plotted as functions of the Sommerfeld number for the cases of centrally loaded for various misalignment conditions.

2 Bearing Lubrication Analysis

An overview of the Reynolds' equation and appropriate boundary conditions for calculating the lubricant film pressure in journal bearings is included in Appendix. The central difference expansions can be used to derive the dimensionless Reynolds' equation without the first-order pressure derivative term and the second-order pressure derivative pressure. Lubricant film stiffness and damping matrices correspond to these equations. These equations are subsequently solved using a finite difference formulation.

2.1 Linearized Characteristics.

This section will evaluate the dynamic behavior of a misaligned journal bearing when a slight amplitude vibration is superimposed on a steady running condition. The force and moment components on the journal under this condition are functions of the coordinates

u_{0}, v_{0}

of the journal center

O_{J} (Z = 0)

and the misalignment of angles

Ψ_{X}, Ψ_{Y}

⁠. The film force components are based in Appendix A.3

\begin{matrix} F_{X A} = F_{X} (u_{0}, v_{0}, 0, 0) \end{matrix}

(1)

\begin{matrix} F_{Y A} = F_{Y} (u_{0}, v_{0}, 0, 0) \end{matrix}

(2)

where $F_{X}$ and $F_{Y}$ are film force on the journal along axes $O X$ and $O Y$ ⁠; $F_{X A}$ and $F_{Y A}$ are film forces on the journal along axes $O X$ and $O Y$ for steady-running point A.

In general, the film force components can be expressed as

\begin{matrix} F_{X} = F_{X} (u_{0}, v_{0}, \dot{u_{0}}, \dot{v_{0}}) \end{matrix}

(3)

\begin{matrix} F_{Y} = F_{Y} (u_{0}, v_{0}, \dot{u_{0}}, \dot{v_{0}}) \end{matrix}

(4)

Under steady-running conditions, the journal has only rotational speed about its stationary axis, which generally is not parallel to the bearing axis. This is expressed mathematically by the fact that

u_{0}, v_{0}, {\dot{u}}_{0}, and \dot{v_{0}}

are zero. In lubrication problems, the direction of the steady load to the bearing is specified, and axis

O Y

is selected as the loading direction. As a result, the force component

F_{X}

along the axis,

O X

does not exist. Once the axis

O Y

is established (angle

α

⁠), a specific misalignment is introduced. In the computer program, for a given eccentricity ratio

ε_{0}

⁠, the angle

φ_{0}

is varied until the component

F_{X}

is zero. The convergence criterion is shown as below [16]

\begin{matrix} | \frac{F_{X}}{F_{Y}} | \leq 0.001 \end{matrix}

(5)

Under the assumption of small displacements from the steady running position, Eqs. and may be written

\begin{matrix} \begin{matrix} Γ = Γ_{A} + {(\frac{\partial Γ}{{\partial u}_{0}})}_{A} δ u_{0} + {(\frac{\partial Γ}{{\partial v}_{0}})}_{A} δ v_{0} + {(\frac{\partial Γ}{{\partial \dot{u}}_{0}})}_{A} δ {\dot{u}}_{0} + {(\frac{\partial Γ}{{\partial \dot{v}}_{0}})}_{A} δ {\dot{v}}_{0} \\ (Γ stands for any of F_{X}, F_{Y}) \end{matrix} \end{matrix}

(6)

The dot is the velocity of the journal center at any

Z

direction along axes

O X

and

O Y

in case of misalignment, as shown in Fig. 11. The incremental dynamic forces due to journal vibration are

\begin{matrix} δ F_{X} = F_{X} - F_{X A} \end{matrix}

(7)

\begin{matrix} δ F_{Y} = F_{Y} - F_{Y A} \end{matrix}

(8)

\begin{matrix} \begin{matrix} δ Γ = {(\frac{\partial Γ}{\partial u_{0}})}_{A} δ u_{0} + {(\frac{\partial Γ}{\partial v_{0}})}_{A} δ v_{0} + {(\frac{\partial Γ}{{\partial \dot{u}}_{0}})}_{A} δ {\dot{u}}_{0} + {(\frac{\partial Γ}{{\partial \dot{v}}_{0}})}_{A} δ {\dot{v}}_{0} \\ (Γ stands for any of F_{X}, F_{Y}) \end{matrix} \end{matrix}

(9)

At this point, a set of eight coefficients is introduced, as shown in Table 1. Using these elastic and damping coefficients, the incremental dynamic forces are expressed as

\begin{matrix} δ F_{X} = - K_{X X} δ u_{0} - K_{X Y} δ v_{0} - Q_{X X} δ {\dot{u}}_{0} - Q_{X Y} δ {\dot{v}}_{0} \end{matrix}

(10)

\begin{matrix} δ F_{Y} = - K_{Y X} δ u_{0} - K_{Y Y} δ v_{0} - Q_{Y X} δ {\dot{u}}_{0} - Q_{Y Y} δ {\dot{v}}_{0} \end{matrix}

(11)

Table 1

Definition of stiffness and damping coefficients

Identification	Force coefficients
Stiffness coefficients (displacement)	$\begin{matrix} K_{X X} = - \frac{\partial F_{X}}{{\partial u}_{0}} & K_{Y X} = - \frac{\partial F_{Y}}{{\partial u}_{0}} \\ K_{X Y} = - \frac{\partial F_{X}}{{\partial v}_{0}} & K_{Y Y} = - \frac{\partial F_{Y}}{{\partial v}_{0}} \end{matrix}$
Damping coefficients (displacement)	$\begin{matrix} Q_{X X} = - \frac{\partial F_{X}}{{\partial \dot{u}}_{0}} & Q_{Y X} = - \frac{\partial F_{Y}}{{\partial \dot{u}}_{0}} \\ Q_{X Y} = - \frac{\partial F_{X}}{{\partial \dot{v}}_{0}} & Q_{Y Y} = - \frac{\partial F_{Y}}{{\partial \dot{v}}_{0}} \end{matrix}$

Identification	Force coefficients
Stiffness coefficients (displacement)	$\begin{matrix} K_{X X} = - \frac{\partial F_{X}}{{\partial u}_{0}} & K_{Y X} = - \frac{\partial F_{Y}}{{\partial u}_{0}} \\ K_{X Y} = - \frac{\partial F_{X}}{{\partial v}_{0}} & K_{Y Y} = - \frac{\partial F_{Y}}{{\partial v}_{0}} \end{matrix}$
Damping coefficients (displacement)	$\begin{matrix} Q_{X X} = - \frac{\partial F_{X}}{{\partial \dot{u}}_{0}} & Q_{Y X} = - \frac{\partial F_{Y}}{{\partial \dot{u}}_{0}} \\ Q_{X Y} = - \frac{\partial F_{X}}{{\partial \dot{v}}_{0}} & Q_{Y Y} = - \frac{\partial F_{Y}}{{\partial \dot{v}}_{0}} \end{matrix}$

2.2 Validation Results.

Based on Sec. 2.1, misaligned journals dynamics coefficients method, a Fortran 90 code was developed. Fortran 90 code flowchart is presented in Figs. 2 and 3.

Fig. 2

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Fortran 90 code main code

Fig. 3

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Fortran 90 code Step 5 and Step 6

Read the input data of the bearing parameters (diameter, length, and radial clearance), journal position for bearing parameters (eccentricity ratio, attitude angle, load angle, and misalignment angle), shaft speed, fluid viscosity, and grid number. The definitions of the symbols are shown in Nomenclature.
Set the coefficients to zero at the beginning. The definitions of the symbols and calculation are shown in Nomenclature and Appendix
Calculate the forces based on Eqs. (3) and (4); the convergence criterion is shown in Eq. (5).
Calculate the stiffness and damping coefficients based on Table 1.
Print out the results of stiffness and damping coefficients.

The validation of the Fortran 90 code is based on the numerical results obtained by Lund and Thomsen [17]. The misalignment is introduced using shaft tilting around the x- and y-axes, as shown in Figs. 2 and 3. The normalized misalignment is defined as $Ψ = C / 2 L$ around the bearing midplane (Fig. 4), where $C$ and $L$ are the bearing radial clearance and length, respectively [18]. The results are presented for a normalized misalignment angle of $Ψ_{X} = 0.5$ and $Ψ_{Y} = 0.5$ ⁠. A partial bearing arc length $β$ is used for a 150-deg bearing loaded in the $X$ -direction. The eccentricity ratio range is from 0.1 to 0.65. The journal bearing is lubricated by SAR30/SAE10W40 at 204 °C, and the dynamic viscosity is 24 mPa·s. The geometric characteristics of the journal bearing used by Lund and Thomsen are summarized in Table 2.

Fig. 4

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Partial, 150-deg bearing with misalignment around bearing midplane axes

Table 2

Lund and Thomsen's journal bearing characteristics [17]

Parameter	Symbol	Value
Length/diameter ratio	L/D	$1 / 2$
Clearance (mm)	C	$0.0254$
Journal speed (rpm)	N	$5000$
Normalized misalignment angle in $X$ - and $Y$ -axes	$Ψ_{X}, Ψ_{Y}$	$0.5$
Bearing arc length $(\deg)$	$β$	$150$

To accurately resolve the flow field in the narrow radial clearance, the $m$ and $n$ are odd integers representing the maximum grid index in θ and $Z$ -direction, respectively. As illustrated in Fig. 5, a grid index number study was carried out to balance the computational cost and accuracy. A different $m$ and $n$ grid index number varying from 3 to 49 are studied, and the journal-bearing misalignment model is based on Table 2. In Appendix A.5, Simpson's 1/3 rule was used as a numerical method for approximating the definite the θ and Z of Eqs. (A54) and (A55) in Appendix A.5. The grid numbers are an even number of intervals (or an odd number of points). The relative error of Fortran90 program dynamic coefficient results and Lund's numerical data are shown in Fig. 5 [17]. Convergence was observed with the number of index numbers (⁠ $m$ and $n$ ⁠) larger than 9. In addition, the difference between the middle grid (i.e., 9) and the finest grids (i.e., 49) is constantly less than 5%. Thus, the grid indexes 9 were used for all Fortran90 program dynamic coefficients simulations.

Fig. 5

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Grid index number of $m, n$ convergence study compared with Lund's data

The data of dynamic coefficients from Lund's numerical results were used to validate the Fortran90 code. The code was validated with numerical results from the reference. The stiffness and damping coefficients with the journal velocity of 5000 rpm between the experimental data and simulation results were compared in Figs. 6 and 7. The Fortran90 code results showed good agreement with Lund and Thomsen's numerical results, and the relative discrepancy was within ±5%.

Fig. 6

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Comparison of dynamic stiffness coefficients at 204 °C for $L / D$ = 0.5

Fig. 7

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Comparison of dynamic damping coefficients at 204 °C for $L / D$ = 0.5

To investigate the dynamic coefficients on the journal bearing with FLiBe at 649 °C was simulated. The bearing parameters and operation circumstances are based on Smith's design [9]. The detailed dimensions of the journal bearings are summarized in Table 3. The FLiBe was used as a lubricant, and the simulated conditions are 649 °C and atmospheric pressure. The results are presented for a normalized misalignment angle of $Ψ_{X} = 0.5$ and $Ψ_{Y} = 0.5$ ⁠. A partial bearing arc length $β$ is used 360-deg bearing loaded in the X-direction. FLiBe lubricates the journal bearing, and the dynamic viscosity is 8 mPa·s.

Table 3

The journal bearing characteristics with FLiBe at 649 °C

Parameter	Symbol	Value
Length/diameter ratio	L/D	$1$
Clearance (mm)	C	$0.127$
Journal speed (rpm)	N	$1200$
Normalized misalignment angle in $X$ - and $Y$ -axes	$Ψ_{X}, Ψ_{Y}$	$0.5$
Bearing arc length $(\deg)$	$β$	$360$

Figures 8 and 9 show that when the bearing is lightly loaded at the eccentricity ratio from 0.1 to 0.3 with FLiBe salt lubricate bearing, the dynamic coefficients are much larger than those at the heavily loaded region. On the other hand, the stable eccentricity ratio region is from 0.3 to 0.6. However, in the high eccentricity ratio region, the dynamic coefficients will have a trend to increase. High radial rotor loads cause significant changes in these patterns: imbalance does not occur when the journal rotates around an eccentric position within the bearing. Journal bearings' cross-coupled stiffness $(K_{X Y}, K_{Y X})$ destabilizes the system. Moreover, such bearing damping $(Q_{X X}, Q_{Y Y})$ is high.

Fig. 8

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Dynamic stiffness coefficients for $L / D$ = 1 with 649 °C FLiBe

Fig. 9

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Dynamic damping coefficients for $L / D$ = 1 with 649 °C FLiBe

3 Conclusions

This study investigated the journal-bearings dynamic characteristics under hydrodynamic lubrication conditions. A Fortran 90 code was developed to evaluate the dynamic coefficients and the code was verified using Lund and Thomeson's numerical results. The code results show good agreement with the numerical results from the reference, and the relative discrepancy is within ±5%. The result of the FLiBe salt journal bearing followed a similar trend and agreed with that of the 150 deg tilting pad bearing with oil lubricants. Dynamic coefficient results for molten salt lubricated journal bearings found considerable 6–11 variation in the dynamic coefficient at light loads (journal eccentricity ratio range 0.1–0.3). The stable journal eccentric ratio region is from 0.3 to 0.6. Therefore, the medium-loaded molten salt lubricated journal bearing shows the most durability in high-temperature molten salt conditions. Finding the stable load region is important for the molten salt-lubricated journal bearing for a long-term stable operation and avoiding catastrophic failure. These dynamic coefficients will be helpful for future experiments and FHR pump-bearing designers.

Funding Data

U.S. Department of Energy Office of Nuclear Energy's Nuclear Energy University Program (Award No. DENE0008977; Funder ID: 10.13039/100006147).

Data Availability Statement

The authors attest that all data for this study are included in the paper.

Nomenclature

$C$ =: radial clearance, mm
$D$ =: bearing diameter, mm
$e$ =: journal center eccentricity, mm
$F$ =: film force on journal, $N$
$h$ =: film thickness, mm
$i$ =: grid index in the $θ$ -direction
$j$ =: grid index in the $Z$ -direction
$K$ =: stiffness coefficient of film force on journal displacement. Coordinate system $OXY$ ⁠, also Table 2
$L$ =: length of bearing in the z-direction, mm
$m$ =: odd integer representing maximum grid index in $θ$ -direction
$n$ =: odd integer representing maximum grid index in $Z$ -direction
$N$ =: journal rotation rate, rpm
$O$ =: bearing center
$p$ =: lubricant pressure, Pa
P =: dimensionless lubrication pressure
$Q$ =: damping coefficient of film force on journal displacement. Coordinate system $OXY$ ⁠, also Table 2
$R$ =: journal radius, mm
$u$ =: coordinate of axis $O X$ of journal center $O_{J}^{'}$ at any $z$ in case of misalignment
$U$ =: velocity of journal surface parallel to film, m/s
$v$ =: coordinate of axis $O Y$ of journal center $O_{J}^{'}$ at any $z$ in case of misalignment
$V$ =: velocity of journal surface normal to film, m/s
$W$ =: journal static load, N
$z$ =: coordinate along bearing axis, mm
$Z$ =: dimensionless axial coordinateness; $Z = h / C$

$α$ =: angle between the leading edge of the bearing and the steady load or axis $O Y,$ deg
$β$ =: bearing arc length, deg
$γ$ =: constant coefficient used in the difference Eq. (A58), defined by Eq. (A59)
$Γ$ =: stands for any of $F_{X}, F_{Y}$
$δ$ =: coefficient used in the difference Eq. (A58), defined by Eq. (A62)
$Δ Z$ =: finite increment in $Z$ -direction
$Δ θ$ =: finite increment in $θ$ -direction
$ε$ =: eccentricity ratio of journal center
$θ$ =: coordinate measured from line of centers, deg
$μ$ =: absolute viscosity, mPa·s
$ξ$ =: angle between the angle of $φ_{0}$ and $φ$ ⁠, deg
$σ$ =: component of the velocity of journal center $O_{J} at (z = 0)$
$τ$ =: angular velocity
$φ$ =: angle to determine the position of the journal center with respect to the bearing. In case of misalignment journal center $O_{J}^{'}$ is considered (Fig. 6), deg
$ϕ$ =: attitude angle, i.e., angle between static load and line of centers, deg
$Ψ$ =: rotation angle, deg
$ω$ =: journal angular velocity, rad/s

$_{\min}$ =: minimum value
$_{X}$ =: components of film force on the journal along axis $O X$
$_{X A}$ =: components of film force on the journal along axis $O X$ for steady-running point A
$_{Y}$ =: components of film force on the journal along axis $O Y$
$_{Y A}$ =: components of film force on the journal along axis $O Y$ for steady-running point A
$_{0}$ =: at plane $z = 0$ in case of misalignment
$_{1}$ =: leading edge
$_{2}$ =: trailing edge
$\dot{}$ =: component of the velocity
$^{*}$ =: dimensionless dependent variable

FHRs =: fluoride-salt-cooled high-temperature reactors
IHX =: an intermediate heat exchanger
MSFR =: molten fast-spectrum reactor
MSR =: molten salt reactor

A.1 Reynolds' Equation.

The Reynolds' equation describes the lubricant film pressure distribution in journal bearings, derived from the Navier–Stokes equations for incompressible flow and the continuity equation under simplifying assumptions. For a Newtonian fluid, the Reynolds' equation is written as Eq. (A1).

The differential equation which governs the pressure

p

in the lubricant film for journal bearings is

\begin{matrix} \frac{1}{R^{2}} \frac{\partial}{\partial θ} (h^{3} \frac{\partial p}{\partial θ}) + \frac{\partial}{\partial z} (h^{3} \frac{\partial p}{\partial z}) = \frac{μ}{R} \frac{\partial h}{\partial θ} (U h) - 12 μ V \end{matrix}

(A1)

where $h$ is the film thickness at a point determined by the coordinates $θ$ and $z$ ⁠, $R$ is the journal radius, and $μ$ is the lubricant viscosity. Furthermore, $U$ and $V$ are the velocity components of the journal surface in the direction of the rotational motion and perpendicular to it, respectively. This equation is valid for incompressible Newtonian lubricant and laminar flow. The derivation of Reynolds' equation (A1) and the associated assumptions are given in Ref. [19]. In Eq. (A1), the fluid is assumed to be isothermal and incompressible with constant dynamic viscosity. The pressure change is negligible in the radial direction because the $h$ is negligibly small.

A.2 Steady Running Conditions.

Under steady-running conditions [20], the journal axis

O^{'}

remains stationary, i.e., (Fig. 10)

\begin{matrix} V = 0 \end{matrix}

(A2)

\begin{matrix} U = ω R \end{matrix}

(A3)

Fig. 10

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Plain cylindrical bearing

At the journal axis

O'

an eccentric position with respect to the stationary element (Fig. 10),

ω

is journal angular velocity. Assuming that the journal axis is parallel to the bearing axis, the film thickness at the point determined by the coordinates

θ

is

\begin{matrix} h = C (1 + ε \cos θ) \end{matrix}

(A4)

where $ε$ is the eccentricity ratio of the journal center, $ε = e / C$ ⁠, $C$ is the clearance between the journal bearing, $e$ is the journal center eccentricity.

Considering a constant oil viscosity throughout the bearing, Reynolds' equation becomes

\begin{matrix} \frac{\partial}{\partial θ} [{(1 + ε \cos θ)}^{3} \frac{\partial p}{\partial θ}] + R^{2} \frac{\partial}{\partial z} [{(1 + ε \cos θ)}^{3} \frac{\partial p}{\partial z}] = - 6 μ ω {(\frac{R}{C})}^{2} ε sin θ \end{matrix}

(A5)

As shown in Fig. 11,

L

is the length of the bearing,

O

and

O^{'}

are the bearing and journal center points,

W

is the journal load, and

ϕ

is the angle between the load and line of centers. The boundary conditions associated with this partial differential equation are

\begin{matrix} p (θ, \frac{L}{2}) = 0 \end{matrix}

(A6)

\begin{matrix} p (θ, - \frac{L}{2}) = 0 \end{matrix}

(A7)

\begin{matrix} p (θ_{1}, z) = 0 \end{matrix}

(A8)

\begin{matrix} p (θ_{2}, z) = 0 \end{matrix}

(A9)

Fig. 11

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The geometry of a journal bearing

where $θ_{1} and θ_{2}$ are coordinated to the leading edge of the bearing and coordinated to the trailing edge of the bearing.

If the negative pressure is developed at the trailing end of the pressure profile, then condition equation should be neglected. Instead, it is usually accurate enough to consider the following two conditions:

\begin{matrix} p (θ_{2}^{*}, z) = 0 \end{matrix}

(A10a)

\begin{matrix} \frac{\partial}{\partial θ} [p (θ_{2}^{*}, z)] = 0 \end{matrix}

(A10b)

where $θ_{2}^{*}$ is dimensionless coordinate to the trailing edge of the bearing.

In a journal bearing with finite $L / D$ ratio, the position where boundary conditions (A10a) and (A10b) occur is not independent of the coordinate $z$ ⁠. $L and D$ are the length and diameter of the bearing, respectively. The line where the pressure gradient disappears is curved, and a fine-mesh grid can be used to determine its shape.

A.3 Geometry With Misalignment.

The behavior of a misaligned journal bearing can be expressed by four stiffness and four damping coefficients. The four stiffness and damping coefficients are the linear terms of a Taylor series expansion of the journal bearing forces and moments when an equilibrium state. The relative position of the journal axis concerning the bearing (Fig. 12(a)) can be determined by the angle $Ψ_{X}, Ψ_{Y}$ (Fig. 12(b)) and coordinates $u_{0}, v_{0}$ of point $O_{J}$ (Fig. 12(c)). $O_{J}$ is the point where the journal axis intersects the $OXY$ plane [21].

Fig. 12

View large Download slide

Parameters necessary to determine the relative position of the journal axis with respect to the bearing (a), (b), (c)

$ψ_{X}$ and $ψ_{Y}$ are the rotation angles about axes $O X$ and $O Y$ ⁠. The original plane is $OXYZ$ ⁠. ${O X}_{1} Y_{1} Z_{1}$ is the position of $OXYZ$ when it is rotated about axis $O X$ by an angle $Ψ_{X}$ ⁠. Furthermore, ${O X}_{2} Y_{2} Z_{2}$ is the position of ${O X}_{1} Y_{1} Z_{1}$ when it is rotated about the axis ${O Y}_{1}$ by an angle $Ψ_{Y}$ ⁠. Finally, the journal axis position is reached by translating ${O X}_{2} Y_{2} Z_{2}$ so that point $O$ coincides with $O_{J}$ ⁠. The positive direction of $Ψ_{X}$ and $Ψ_{Y}$ is shown in Fig. 12(b) [21].

The coordinates

X_{2}, Y_{2}, Z_{2}

of any point concerning

{O X}_{2} Y_{2} Z_{2}

and the coordinates

X, Y, Z

of the same point concerning

OXYZ

⁠, are related as follows:

\begin{matrix} [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 & 0 & Ψ_{Y} \\ 0 & 1 & Ψ_{X} \\ - Ψ_{Y} & - Ψ_{X} & 1 \end{matrix}] [\begin{matrix} X_{2} \\ Y_{2} \\ Z_{2} \end{matrix}] \end{matrix}

(A11)

Considering now any point on the axis

{O Z}_{2}

(i.e.,

X_{2} = 0, Y_{2} = 0

⁠) the above matrix equation yields

\begin{matrix} x = Z_{2} Ψ_{Y} \end{matrix}

(A12)

\begin{matrix} y = Z_{2} Ψ_{X} \end{matrix}

(A13)

\begin{matrix} z = Z_{2} \end{matrix}

(A14)

The coordinates

u

(axis

O X

⁠) and

v

(axis

O Y

⁠) of any point located on the journal axis are

\begin{matrix} u = u_{0} + Z Ψ_{Y} \end{matrix}

(A15)

\begin{matrix} v = v_{0} + Z Ψ_{X} \end{matrix}

(A16)

The position of the point

O_{J}

may be determined either by the coordinates

u_{0}, v_{0}

or by the pair of polar coordinates

e_{0}, ϕ_{0}

⁠, because

\begin{matrix} u_{0} = e_{0} sin φ_{0} \end{matrix}

(A17)

\begin{matrix} v_{0} = e_{0} cos φ_{0} \end{matrix}

(A18)

The film thickness

h

at any location

z

of the bearing and any angular position

θ

is given by

\begin{matrix} h (θ, Z) = C + ecos (θ - ξ) \end{matrix}

(A19)

ξ

is the angle between the angle

φ_{0}

to determine the position of the original journal center

O_{J}

and the angle

φ

to determine the misalignment journal center

O_{J}^{'}

\begin{matrix} ξ = φ - φ_{0} \end{matrix}

(A20)

Substituting this value of

ξ

in Eq. yields

\begin{matrix} h = C + e \cdot \cos (θ + φ_{0} - φ) \end{matrix}

(A21)

or

\begin{matrix} h = C + e \cdot cos φ \cdot cos (θ + φ_{0}) + e \cdot sin φ \cdot sin (θ + φ_{0}) \end{matrix}

(A22)

Combining Eqs. and with the above, it follows that:

\begin{matrix} h = C + u \cdot sin (θ + φ_{0}) + v \cdot cos (θ + φ_{0}) \end{matrix}

(A23)

Substituting in this equation the values of

u

and

v

from Eqs. and and considering Eqs. and , it is found that

\begin{matrix} h = C + e_{0} cos θ + z [Ψ_{X} c os (θ + φ_{0}) + Ψ_{Y} s in (θ + φ_{0})] \end{matrix}

(A24)

Thus, the film thickness $h$ is a linear function of $z$ ⁠, $Ψ_{X}$ ⁠, and $Ψ_{Y}$ ⁠.

Another useful parameter is the minimum value of the film thickness $h_{\min}$ ⁠, which occur at one of the two ends of the bearing.

It can be shown that this is

\begin{matrix} h_{\min} (z = \frac{L}{2}) = C - \sqrt{{(e_{0} s in ϕ_{0} + Ψ_{Y} L / 2)}^{2} + {(e_{0} c os ϕ_{0} + Ψ_{X} L / 2)}^{2}} \end{matrix}

(A25a)

\begin{matrix} h_{\min} (z = - \frac{L}{2}) = C - \sqrt{{(e_{0} s in ϕ_{0} - Ψ_{Y} L / 2)}^{2} + {(e_{0} c os ϕ_{0} - Ψ_{X} L / 2)}^{2}} \end{matrix}

(A25b)

where

h

was already found and given by Eq. , i.e.,

ϕ_{0}

attitude angle for steady-running

\begin{matrix} \frac{\partial h}{\partial θ} = - e_{0} sin θ + z [- Ψ_{X} s in (θ + φ_{0}) + Ψ_{X} c os (θ + φ_{0})] \end{matrix}

(A26)

\begin{matrix} - V = {\dot{u}}_{0} sin (θ + φ_{0}) + {\dot{v}}_{0} cos (θ + φ_{0}) + z [{\dot{Ψ}}_{X} c os (θ + φ_{0}) + {\dot{Ψ}}_{Y} s in (θ + φ_{0})] \end{matrix}

(A27)

A.4 Reynolds' Equation With Misalignment.

To write Reynolds' equation in dimensionless form, the following dimensionless quantities are defined:

\begin{matrix} ε = \frac{e}{C} \end{matrix}

(A28)

\begin{matrix} ε_{0} = \frac{e_{0}}{C} \end{matrix}

(A29)

\begin{matrix} H = \frac{h}{C} \end{matrix}

(A30)

\begin{matrix} Z = \frac{z}{L} \end{matrix}

(A31)

\begin{matrix} P = \frac{p}{ω μ} {(\frac{C}{R})}^{2} \end{matrix}

(A32)

\begin{matrix} U^{*} = \frac{U}{R ω} \end{matrix}

(A33)

\begin{matrix} V^{*} = \frac{V}{C ω} \end{matrix}

(A34)

\begin{matrix} σ_{X}^{*} = \frac{{\dot{u}}_{0}}{C ω} \end{matrix}

(A35)

\begin{matrix} σ_{Y}^{*} = \frac{{\dot{v}}_{0}}{C ω} \end{matrix}

(A36)

\begin{matrix} Ψ_{X}^{*} = \frac{Ψ_{X} L}{C} \end{matrix}

(A37)

\begin{matrix} Ψ_{Y}^{*} = \frac{Ψ_{Y} L}{C} \end{matrix}

(A38)

\begin{matrix} τ_{X}^{*} = \frac{{\dot{Ψ}}_{X} L}{C ω} \end{matrix}

(A39)

\begin{matrix} τ_{Y}^{*} = \frac{{\dot{Ψ}}_{Y} L}{C ω} \end{matrix}

(A40)

Substituting Eqs. – into Eqs. and , , and , the following dimensionless form was obtained for Reynolds' equation:

\begin{matrix} \frac{\partial}{\partial θ} (H^{3} \frac{\partial P}{\partial θ}) + \frac{D^{2}}{{4 L}^{2}} \frac{\partial}{\partial Z} (H^{3} \frac{\partial P}{\partial Z}) = 6 \frac{\partial H}{\partial θ} - 12 V^{*} \end{matrix}

(A41)

where

\begin{matrix} H = 1 + ε_{0} cos θ + Z [Ψ_{X} c os (θ + φ_{0}) + Ψ_{Y} s in (θ + φ_{0})] \end{matrix}

(A42)

\begin{matrix} \frac{\partial H}{\partial θ} = - ε_{0} sin θ + Z [- Ψ_{X}^{*} s in (θ + φ_{0}) + Ψ_{Y}^{*} c os (θ + φ_{0})] \end{matrix}

(A43)

\begin{matrix} - V^{*} = σ_{X}^{*} sin (θ + φ_{0}) + σ_{Y}^{*} cos (θ + φ_{0}) + Z [τ_{X}^{*} c os (θ + φ_{0}) + τ_{Y}^{*} s in (θ + φ_{0})] \end{matrix}

(A44)

Defining

\begin{matrix} F (θ, Z) = 6 \frac{\partial H}{\partial θ} - 12 V^{*} \end{matrix}

(A45)

Equation is written as

\begin{matrix} \frac{\partial}{\partial θ} (H^{3} \frac{\partial P}{\partial θ}) + \frac{D^{2}}{{4 L}^{2}} \frac{\partial}{\partial Z} (H^{3} \frac{\partial P}{\partial Z}) = F (θ, Z) \end{matrix}

(A46)

The boundary condition is now written in dimensionless form as follows: Eqs. (A10a) and (A10b).

A.5 The Difference Equation.

A dimensionless Reynolds' equation without a first-order derivative term can be obtained using the following new dependent variable [22]:

\begin{matrix} P^{*} = P {\cdot H}^{\frac{3}{2}} \end{matrix}

(A47)

with which a more accessible and more accurate numerical evaluation of the pressure distribution may be made.

On making the substitution , Reynolds' equation in dimensionless form (i.e., Eq. ) becomes

\begin{matrix} \frac{\partial^{2} P^{*}}{{\partial θ}^{2}} + {(\frac{D}{2 L})}^{2} \frac{\partial^{2} P^{*}}{{\partial Z}^{2}} - \frac{3}{4} \frac{P^{*}}{H^{2}} [{(\frac{\partial H}{\partial θ})}^{2} + 2 H \frac{\partial^{2} H}{{\partial θ}^{2}} + {(\frac{D}{2 L})}^{2} {(\frac{\partial H}{\partial Z})}^{2} + 2 H {(\frac{D}{2 L})}^{2} \frac{\partial^{2} H}{{\partial Z}^{2}}] = H^{- \frac{3}{2}} \cdot F (θ, Z) \end{matrix}

(A48)

In the case under consideration,

H

is given by Eq. and is a linear function of

Z

⁠, which means that

\begin{matrix} \frac{\partial^{2} H}{{\partial Z}^{2}} = 0 \end{matrix}

(A49)

Under this condition, Reynolds' equation is written as

\begin{matrix} \frac{\partial^{2} P^{*}}{{\partial θ}^{2}} + {(\frac{D}{2 L})}^{2} \frac{\partial^{2} P^{*}}{{\partial Z}^{2}} - \frac{3}{4} \frac{P^{*}}{H^{2}} [{(\frac{\partial H}{\partial θ})}^{2} + 2 H \frac{\partial^{2} H}{{\partial θ}^{2}} + {(\frac{D}{2 L})}^{2} {(\frac{\partial H}{\partial Z})}^{2}] = H^{- \frac{3}{2}} \cdot F (θ, Z) \end{matrix}

(A50)

In the equation above, the term

H

⁠,

\partial H / \partial θ,

and

F (θ, Z)

have been derived in the Appendix A.4 and A.5. Some additional terms are given below

\begin{matrix} \frac{\partial^{2} H}{{\partial θ}^{2}} = - ε_{0} cos θ + Z [- Ψ_{X}^{*} c os (θ + φ_{0}) - Ψ_{Y}^{*} s in (θ + φ_{0})] \end{matrix}

(A51)

Comparing Eq. with Eq. yields

\begin{matrix} \frac{\partial^{2} H}{{\partial θ}^{2}} = 1 - H \end{matrix}

(A52)

Differentiating Eq. concerning

Z

yields

\begin{matrix} \frac{\partial H}{\partial Z} = Ψ_{X}^{*} cos (θ + φ_{0}) + Ψ_{Y}^{*} sin (θ + φ_{0}) \end{matrix}

(A53)

To solve Eq. (A51) numerically, the bearing surface is “unwrapped” onto a plane and is covered with families of lines parallel to the $θ$ and $Z$ axes, respectively. This grid is shown in Fig. 13. The derivatives of $P^{*}$ are expressed in terms of three-point central differences [16], yielding and algebraic equations relating $P_{i, j}^{*}$ ⁠, and its four adjacent values, with coefficients known depending on the film thickness. The object of the present section is to develop this differential equation.

Fig. 13

View large Download slide

Finite difference grid

The second-order derivatives of

P^{*}

concerning

θ

and

Z

may be approximated by the following central difference expansions:

\begin{matrix} \frac{\partial^{2} P^{*}}{{\partial θ}^{2}} = {(\frac{1}{Δ θ})}^{2} [P_{i - 1, j}^{*} - 2 P_{i, j}^{*} + P_{i + 1, j}^{*}] \end{matrix}

(A54)

\begin{matrix} \frac{\partial^{2} P^{*}}{{\partial Z}^{2}} = {(\frac{1}{Δ Z})}^{2} [P_{i, j - 1}^{*} - 2 P_{i, j}^{*} + P_{i, j + 1}^{*}] \end{matrix}

(A55)

where

\begin{matrix} Δ θ = \frac{β}{(m + 1)} \end{matrix}

(A56)

\begin{matrix} Δ Z = \frac{L}{(n + 1)} \end{matrix}

(A57)

In the equations above, $P_{i, j}^{*}$ is the value of $P^{*}$ at point $i, j$ of the grid.

Substituting the expressions above for the second-order derivatives into Eq. yields this equation's finite difference form. It is not difficult to prove that

\begin{matrix} P_{i - 1, j}^{*} + {γ P}_{i, j - 1}^{*} + δ_{i, j} P_{i, j}^{*} + {γ P}_{i, j + 1}^{*} + P_{i + 1, j}^{*} = ξ_{i, j} \end{matrix}

(A58)

where

\begin{matrix} γ = {(\frac{D}{2 L})}^{2} {(\frac{Δ θ}{Δ Z})}^{2} \end{matrix}

(A59)

Furthermore, coefficients

δ_{i, j}

⁠,

ξ_{i, j}

are calculated at the points

i, j,

which has the following coordinates:

\begin{matrix} θ_{i} = θ_{1} + i (Δ θ) \end{matrix}

(A60)

\begin{matrix} Z_{i} = j (Z) - 0.5 \end{matrix}

(A61)

and they are given by the following expressions:

\begin{matrix} δ_{i, j} = - {2 (1 + γ) + \frac{3}{4} \frac{Δ θ^{2}}{H^{2}} [{(\frac{\partial H}{\partial θ})}^{2} + 2 H \frac{\partial^{2} H}{{\partial θ}^{2}} + {(\frac{D}{2 L})}^{2} {(\frac{\partial H}{\partial Z})}^{2}]} \end{matrix}

(A62)

\begin{matrix} ξ_{i, j} = {(Δ θ)}^{2} H^{- \frac{3}{2}} F (θ, Z) \end{matrix}

(A63)

Analytical expressions for $H$ ⁠, $\partial H / \partial θ, \partial^{2} H / {\partial θ}^{2}$ ⁠, and $\partial H / \partial Z$ are given by Eqs. (A42), (A43), (A52), and (A53), respectively.

After calculating the pressure distribution, double numerical integration can determine the bearing load and moment components. For this purpose, the Simpson's 1/3 rule is applied [16], requiring that M and N be odd integers.

References

1.

Rosenthal

,

M. W.

,

Kasten

,

P. R.

, and

Briggs

,

R. B.

,

1970

, “

Molten-Salt Reactors—History, Status, and Potential

,”

Nucl. Appl. Technol.

,

8

(

2

), pp.

107

–

117

.10.13182/NT70-A28619

Google Scholar

Crossref

2.

Holcomb

,

D. E.

,

Flanagan

,

G. F.

,

Mays

,

G. T.

,

Pointer

,

W. D.

,

Robb

,

K. R.

, and

Yoder

,

G. L.

, Jr. ,

2013

, “

Fluoride Salt-Cooled High-Temperature Reactor Technology Development and Demonstration Roadmap

,” ORNL, Oak Ridge, TN, Report No. ORNL/TM-2013/401.

3.

Renault

,

P.

,

2014

, “

Approach and Challenges for the Seismic Hazard Assessment of Nuclear Power Plants: The Swiss Experience

,”

Boll. Geofis. Teor. Appl.

,

55

(

1

), pp.

149

–

164

.10.4430/bgta0089

4.

Stempien

,

J. D.

,

Ballinger

,

R. G.

, and

Forsberg

,

C. W.

,

2016

, “

An Integrated Model of Tritium Transport and Corrosion in Fluoride Salt-Cooled High-Temperature Reactors (FHRs)–Part I: Theory and Benchmarking

,”

Nucl. Eng. Des.

,

310

, pp.

258

–

272

.10.1016/j.nucengdes.2016.10.051

Google Scholar

Crossref

5.

Cerullo

,

N.

, and

Lomonaco

,

G.

, IV ,

2012

, “

Generation IV Reactor Designs, Operation and Fuel Cycle

,”

Nuclear Fuel Cycle Science and Engineering

,

Elsevier

, Sawston, Cambridge, UK, pp.

333

–

395

.

Google Scholar

Crossref

6.

Terbish

,

J.

, and

van Rooijen

,

W. F. G.

,

2021

, “

Design and Neutronic Analysis of the Intermediate Heat Exchanger of a Fast-Spectrum Molten Salt Reactor

,”

Nucl. Eng. Technol.

,

53

(

7

), pp.

2126

–

2132

.10.1016/j.net.2021.01.021

Google Scholar

Crossref

7.

Di Ronco

,

A.

,

Cammi

,

A.

, and

Lorenzi

,

S.

,

2020

, “

Preliminary Analysis and Design of the Heat Exchangers for the Molten Salt Fast Reactor

,”

Nucl. Eng. Technol.

,

52

(

1

), pp.

51

–

58

.10.1016/j.net.2019.07.013

Google Scholar

Crossref

8.

Liu

,

Y.

,

Che

,

S.

,

Burak

,

A.

,

Barth

,

D. L.

,

Zweibaum

,

N.

, and

Chen

,

M.

,

2023

, “

Numerical Investigations of Molten Salt Pump Journal Bearings Under Hydrodynamic Lubrication Conditions for FHRs

,”

Nucl. Sci. Eng.

,

197

(

5

), pp.

907

–

919

.10.1080/00295639.2022.2103343

Google Scholar

Crossref

9.

Smith

,

P. G.

,

1961

, “

High-Temperature Molten-Salt Lubricated Hydrodynamic Journal Bearings

,”

ASLE Trans.

,

4

(

2

), pp.

263

–

274

.10.1080/05698196108972438

Google Scholar

Crossref

10.

Scarlat

,

R. O.

, and

Peterson

,

P. F.

,

2014

, “

The Current Status of Fluoride Salt Cooled High Temperature Reactor (FHR) Technology and Its Overlap With HIF Target Chamber Concepts

,”

Nucl. Instrum. Methods Phys. Res., Sect. A

,

733

, pp.

57

–

64

.10.1016/j.nima.2013.05.094

Google Scholar

Crossref

11.

Smith

,

P. G.

,

1967

, “

Experience With High-Temperature Centrifugal Pumps in Nuclear Reactors and Their Application to Molten-Salt Thermal Breeder Reactors

,”

Oak Ridge National Laboratory (ORNL)

, Oak Ridge, TN, Report No. ORNL-TM-1993.

12.

Muszynska

,

A.

,

2005

,

Rotordynamics

,

CRC Press

,

Boco Raton, FL

.

Google Scholar

Crossref

13.

Childs

,

D.

, and

Hale

,

K.

,

1994

, “

A Test Apparatus and Facility to Identify the Rotordynamic Coefficients of High-Speed Hydrostatic Bearings

,”

ASME J. Tribol.

,

116

(

2

), pp.

337

–

343

.10.1115/1.2927226

Google Scholar

Crossref

14.

Bompos

,

D. A.

, and

Nikolakopoulos

,

P. G.

,

2016

, “

Rotordynamic Analysis of a Shaft Using Magnetorheological and Nanomagnetorheological Fluid Journal Bearings

,”

Tribol. Trans.

,

59

(

1

), pp.

108

–

118

.http://www.tribology.fink.rs/journals/2015/2015-3/9.pdf

Google Scholar

Crossref

15.

Wang

,

Q. J.

, and

Chung

,

Y. W.

,

2013

,

Encyclopedia of Tribology

,

Springer

,

New York

.

Google Scholar

Crossref

16.

Salvadori

,

M. G.

, and

Baron

,

M. L.

,

1961

,

Numerical Methods in Engineering

(Prentice-Hall Civil Engineering and Engineering Mechanics Series),

Prentice Hall

,

Englewood Cliffs, NJ

.

17.

Lund

,

J. W.

, and

Thomsen

,

K. K.

,

1978

, “

A Calculation Method and Data for the Dynamic Coefficients of Oil-Lubricated Journal Bearings

,”

Topics in Fluid Film Bearing and Rotor Bearing System Design and Optimization

,

Department of Machine Elements, The Technical University of Denmark

,

Lyngby, Denmark

.

18.

Ebrat

,

O.

,

Mourelatos

,

Z. P.

,

Vlahopoulos

,

N.

, and

Vaidyanathan

,

K.

,

2004

, “

Calculation of Journal Bearing Dynamic Characteristics Including Journal Misalignment and Bearing Structural Deformation©

,”

Tribol. Trans.

,

47

(

1

), pp.

94

–

102

.10.1080/05698190490278994

Google Scholar

Crossref

19.

Pinkus

,

O.

,

Sternlicht

,

B.

, and

Saibel

,

E.

,

1962

, “

Theory of Hydrodynamic Lubrication

,”

J. Appl. Mech.

20.

Cameron

,

A.

, and

Wood

,

W. L.

,

1949

, “

The Full Journal Bearing

,”

Proc. Inst. Mech. Eng.

,

161

(

1

), pp.

59

–

72

.10.1243/PIME_PROC_1949_161_010_02

Google Scholar

Crossref

21.

Pafelias

,

T. A.

,

1974

, “

Solution of Certain Problems of Viscous Laminar Flow With Application in Engineering Problems

,”

Rensselaer Polytechnic Institute

,

Troy, NY

.

22.

Sassenfeld

,

H.

, and

Walther

,

A.

,

1954

, “

Gleitlagerberchnungen

,”

V. D. I Forschungsheft

, Berlin, p.

441

.

Molten Salt Pump Journal-Bearings Dynamic Characteristics Under Hydrodynamic Lubrication Conditions

Abstract

1 Introduction

2 Bearing Lubrication Analysis

2.1 Linearized Characteristics.

2.2 Validation Results.

3 Conclusions

Funding Data

Data Availability Statement

Nomenclature

A.1 Reynolds' Equation.

A.2 Steady Running Conditions.

A.3 Geometry With Misalignment.

A.4 Reynolds' Equation With Misalignment.

A.5 The Difference Equation.

References

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Molten Salt Pump Journal-Bearings Dynamic Characteristics Under Hydrodynamic Lubrication Conditions

Abstract

1 Introduction

2 Bearing Lubrication Analysis

2.1 Linearized Characteristics.

2.2 Validation Results.

3 Conclusions

Funding Data

Data Availability Statement

Nomenclature

A.1 Reynolds' Equation.

A.2 Steady Running Conditions.

A.3 Geometry With Misalignment.

A.4 Reynolds' Equation With Misalignment.

A.5 The Difference Equation.

References

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