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EPJ Nuclear Sci. Technol. 5, 13 (2019) © C. Tripodo et al., published by EDP Sciences, 2019 https://doi.org/10.1051/epjn/2019029 Nuclear Sciences & Technologies Available online at: https://www.epj-n.org REGULAR ARTICLE Development of a control-oriented power plant simulator for the molten salt fast reactor Claudio Tripodo, Andrea Di Ronco, Stefano Lorenzi, and Antonio Cammi* Politecnico di Milano, Department of Energy, Nuclear Engineering Division, Via La Masa 34, 20156 Milan, Italy Received: 9 April 2019 / Received in final form: 27 July 2019 / Accepted: 26 August 2019 Abstract. In this paper, modelling and simulation of a control-oriented plant-dynamics tool for the molten salt fast reactor (MSFR) is presented. The objective was to develop a simulation tool aimed at investigating the plant response to standard control transients, in order to support the system design finalization and the definition of control strategies. The simulator was developed employing the well tested, flexible and open-source objectoriented Modelica language. A one-dimensional modelling approach was used for thermal-hydraulics and heat transfer. Standard and validated thermal-hydraulic Modelica libraries were employed for various plant components (tubes, pumps, turbines, etc.). An effort was spent in developing a new MSR library modelling the 1D flow of a liquid nuclear fuel, including an ad-hoc neutron-kinetics model which properly takes into consideration the motion of the Delayed Neutron Precursors along the fuel circuit and the consequent reactivity insertion due to the variation of the effective delayed fractions. An analytical steady-state 2-D model of the core and the fuel circuit was developed using MATLAB in order to validate the Decay Neutron Precursors model implemented in the plant simulator. The plant simulator was then employed to investigate the plant dynamics in response to three transients (variation of fuel flow rate, intermediate flow rate and turbine gas flow rate) that are relevant to control purposes. Simulation outcomes highlight the typical reactor-follows-turbine behavior of the MSFR, and they show the small influence of fuel and intermediate flow rate on the reactor power and their strong effects on the temperatures in their respective circuits. Starting from the insights on the reactor behavior gained from the analysis of its free dynamics, the plant simulator here developed will provide a valuable tool in support to the finalization of the design phase, the definition of control strategies and the identification of controlled operational procedures for reactor startup and shutdown. 1 Introduction The objective of this work was to develop a fast-running, control-oriented plant-dynamics simulation tool for the molten salt fast reactor (MSFR) and the associated Balance of Plant, and to use it to investigate and analyze the plant dynamics. The MSFR is the circulating-fuel fast-neutron-spectrum reactor concept currently object of research under the EU SAMOFAR project (http://samofar.eu/), within the international framework for the development of fourthgeneration nuclear reactors known as Generation-IV International Forum [1]. The demonstration of the loadfollowing capabilities and the control operability of the reactor is one of the objectives of the SAMOFAR project. In this view, it is important to rely on a power plant simulator to study the system dynamics and to develop and test the control strategies. Due to the dynamic and control * e-mail: antonio.cammi@polimi.it related purposes of the power plant simulator, an objectoriented modelling approach is selected as suitable choice for the model-based control design. Due to its features in terms of hierarchical structure, abstraction and encapsulation, this approach allows developing a model that satisfies the requirements of modularity, openness and efficiency [2]. A viable path to achieve the above-mentioned goals is constituted by the adoption of the Modelica language [3]. Modelica is an object-oriented, declarative, equation-based language developed for the componentoriented modelling of complex physical and engineering systems [2]. It allows a description of single system components (or objects) directly in terms of physical equations and principles, and to connect different components through standardized interfaces (or connectors). In addition, his acausal component-based modelling strategy provides high reusability of the models and flexibility of the plant configuration, as well as a more realistic description of the plant, since several validated libraries of power plant components exist (e.g. the ThermoPower library [4]). Modelica is open-source and it has already been This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) Fig. 1. MSFR plant conceptual scheme. successfully adopted in different fields, such as automotive, robotics, thermo-hydraulic and mechatronic systems, but also in nuclear simulation field. Plant simulators were developed for control purposes for the ALFRED (Advanced Lead Fast Reactor European Demonstrator) reactor [5] and the IRIS reactor [6]. As simulation environment, Dymola (Dynamic Modelling Laboratory) [7] was adopted, even if open-source implementation can be considered as alternative option, e.g. OpenModelica [8]. In developing the power plant simulator for the MSFR, it is essential to consider the peculiar features of this reactor, firstly the presence of a liquid circulating nuclear fuel that acts contemporarily as coolant. The strong physical coupling of thermo-fluid-dynamics and neutronics which characterizes the MSFR indeed required to take into account the motion of the Delayed Neutron Precursors (DNPs), which circulate along the fuel circuit. A onedimensional modelling approach was therefore employed for the reactor (as well as for the remaining of the plant) as the best compromise between, on one hand, the need to consider the spatial dependence of the DNPs concentration, and, on the other hand, the need to have a computationally efficient, fast running simulation tool suitable to be employed for plant dynamics investigation and subsequently in support to the design of the plant control system. An ad-hoc point-kinetics model, which is able to take into account the DNPs position in the core, was implemented using a hybrid 0D-1D approach. To verify the DNPs model employed in the MSFR power plant simulator and the corresponding predicted values of the effective delayed neutron fractions for the various delayed groups, an analytical steady-state 2-D model®of the reactor core was developed by using the MATLAB software [9], under suitable simplifying assumptions. The plant simulator was then employed to investigate the plant free dynamics (i.e., the plant response with no control actions) in response to different transients that are relevant for the development of the control strategy. Four different transients were simulated and analyzed: (i) a reduction of the fuel mass flow rate; (ii) a reduction of the intermediate salt mass flow rate; (iii) an increase of the helium mass flow rate in the turbine unit; and (iv) an external reactivity insertion. These transients were selected since they involved three of the possible control variables that can be chosen in the control strategy of the reactor for the full power mode, i.e., the operational mode from 50% to 110% of the power. The possibility to control the reactor in this operational mode acting only on the mass flow rates of the different circuit is relevant since the MSFR does not foresee the use of control rods for load-following operation. The paper is organized as follows. In Section 2, the MSFR reference design is briefly presented, whereas the modelling approach employed for the description of the reactor and the Balance of Plant is described in Section 3. Section 4 illustrates an analytical 2-D benchmark model and its results compared with those of the simulator, in Section 5 the simulator is used to investigate the MSFR plant free dynamics, and in Section 6 some conclusions are drawn. 2 Reference plant and reactor description The conceptual scheme of the MSFR BoP is shown in Figure 1. As it can be seen from the figure, the non-nuclear part of the plant consists of a conventional circuit with two loops in series: (i) the Intermediate Loop, through which a fluoride-based molten salt circulates, serves to extract the heat generated in the reactor
through an Intermediate Heat Exchanger (IHX) [10]
and to transport it to the power conversion system; (ii) the Power Conversion Loop, which consists of a conventional Joule-Brayton gas-turbine cycle [11]. 2.1 Reactor fuel circuit and core The main conceptual feature that distinguish the MSFR is the nuclear fuel that is dissolved in a liquid fluoride- or chloride-based salt which acts contemporarily as fuel and coolant. The reference MSFR design [12] is a 3000 MWth reactor with a total fuel salt volume of 18 m3, operated at a mean fuel temperature of 700 °C. The reactor schemes are shown in Figures 2 and 3. The fuel circuit, defined as the circuit containing the fuel salt during power generation, includes the core cavity and the recirculation loops (also C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) Fig. 2. MSFR fuel circuit conceptual scheme. 3 to operate as a waste-burner for transuranic waste produced in traditional once-through nuclear reactors, thereby allowing a significant reduction in radiotoxicity [13]. The liquid nature of the fuel allows adjusting on-line the fissile content, with the consequence that no excess reactivity is required in the core at any time to compensate for temperature and power defects, or to compensate fission-products-related reactivity losses. This means that neither burnable poisons nor long-term-adjustment control rods are needed in the core. The continuous removal of fission products allows a better chemical control and allows removing any FPs-related negative reactivity. In particular, the removal of the main nuclear poison Xenon eliminates the reactor dead-time following shutdowns or power reductions, paving the way to much more flexible reactor operation and load-following applications. Great advantages are also present looking at the intrinsic safety aspects of the MSFR. Since the fuel is in a fluid state, the core meltdown scenario is eliminated by-design and no limits exist for the attainable fuel burnup with respect to rods cladding damage and fission gas release. The low vapor tension of the molten salt allows operating the reactor at atmospheric pressure, reducing mechanical stresses on structural components and excluding all high-pressurerelated accidental scenarios. Besides, in case of accidents an emergency fuel-draining system allows to automatically and passively drain the whole fuel content of the reactor, to assure its sub-criticality, and to passively cool it long-term [14]. Finally, the dual fuel/coolant role of the salt, together with its neutronics characteristics, implies that the MSFR has very large, negative, prompt temperature and void reactivity feedback coefficients, making the reactor extremely stable [15]. 3 MSFR plant simulator modelling Fig. 3. MSFR fuel circuit layout. called ‘sectors’) including the inlet and outlet pipes, a gas injection system, salt-bubble separators, pumps and fuel heat exchangers. Sixteen cooling sectors are arranged circumferentially around the vessel. Due to the liquid nature of the nuclear fuel, which does not require the presence of any solid fuel-element, and the fast neutron spectrum which does not require any moderating materials, the MSFR core is constituted by a simple, empty cavity, surrounded by an axial reflector and a radial blanket. The fuel salt, with an inlet temperature of about 650 °C, enters radially from the bottom into the active zone, where it temporarily reaches criticality and its heated to the outlet temperature of about 750 °C. The fuel then exits from the top of the core and it is recirculated through the 16 fuel sectors. 2.2 MSFR potentialities Thanks to its peculiar features, the MSFR presents numerous advantages that make it attractive for the long-term perspective of the nuclear energy. It can operate with very flexible fuel-cycle strategies, reaching high breeding ratios with the thorium cycle, and it is capable In the perspective of identifying effective plant control strategies for an innovative reactor concept like the MSFR, an essential preliminary step was to acquire sufficiently accurate knowledge and understanding of both the reactor system dynamics and the whole Balance of Plant dynamics. To this aim, a control-oriented plant-dynamics simulator was developed and then used to study the MSFR dynamics. A proper dynamic simulation tool for control-oriented purposes, especially in a preliminary design phase, should satisfy some basic requirements [4,5]. In particular it should be – modular and extensible, in order to be easily modified and updated to follow the design evolutions; – readable, to allow an easy understanding of the equations implemented; – computationally efficient, to allow fast-running and realtime simulations; – be easily integrable with the control system model. With the above requirements to be fulfilled, the modelling choice fell on the Modelica language [3]. Modelica is an object-oriented, acausal, equation-based language which offers great advantages in terms of modularity, extensibility, readability and integrability with control-dedicated software (e.g. MATLAB control 4 C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) Fig. 4. Conceptual scheme adopted for the MSFR neutron-kinetics. toolbox). The simulator was implemented within the Dymola simulation environment [7], which is equipped with state-of-the-art implicit numerical integration algorithms (e.g. DASSL) to handle non-linear differential-algebraic equations sets and with effective homotopy-based model-initialization algorithms [16], and which provides powerful model-linearization tools potentially useful in the future control system design phase. The tested and validated ThermoPower thermal-hydraulic Modelica library [4] has been used for the simulator modelling, and it has been significantly modified and extended into an MSR library to account for all the balance equations pertaining the various nuclear variables (see Sect. 3.1). 3.1 Fuel circuit and core The usual approach employed for dynamics and control in conventional solid-fueled reactors is the so-called PointKinetics (i.e., zero-dimensional kinetics) [5,6]. In a circulating-fuel reactor like the MSFR the DNPs move along the fuel circuit, and a proper neutronics modelling needed to take into account the position of emission of the delayed neutrons in the core. Besides, a fraction of the delayed neutrons are emitted in the out-of-core portion of the primary circuit, thereby reducing the effective delayed neutron fraction beff [17], with a clear impact on the reactor dynamics. An ad-hoc neutronics model, which is able to take into account the DNPs position in the core, was therefore developed using a hybrid 0D-1D approach. Similar approaches have been proposed in previous works on circulating-fuel reactors’ dynamics [18]. The conceptual scheme adopted for the fuel circuit modelling is shown in Figure 4. The circuit thermal-hydraulics determines the spatial distribution of the DNPs concentration along the fuel circuit. The DNPs spatial profile is then used to compute an effective core-averaged value of the DNPs concentration in the core, suitable to be used in the reactor kinetics equation [19]. To correctly account for the drift of the DNPs, i.e., the fact that they are created in a different location with respect to the emission of the corresponding delayed neutron, in the averaging procedure the delayed neutron source intensity can be weighted with a neutronimportance function that can be both the direct flux or more properly the adjoint neutron flux [20]. Similarly, the average temperature used for the feedback reactivity evaluation is obtained as weighted-average of the temperature profile in the core multiplied by the importance function. The decay heat distribution was modelled using the same 1-D modelling approach. The total reactor power is the sum of the fission power in the core and the decay power throughout the whole fuel circuit. The Modelica model of the fuel circuit is shown in Figure 5. The thermal-hydraulics of the reactor core was modelled in the MSR_Core component (Fig. 5). It is described by the mass (Eq. (1)), X-momentum (Eq. (2)), energy (Eq. (3)) conservation equation and the balance for the DNPs concentration for the 8 DNPs groups (Eq. (4)) and Decay Heat (DH) concentration for the 3 decay-heat groups (Eq. (5)). In the last three equations, the generation term due to the fission process is included. Longitudinal heat and species diffusion were neglected. A ∂d ∂w þ ¼0 ∂t ∂x ð1Þ C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) The time evolution of the normalized core fission power nfiss(t) = Qfiss(t)/Qfiss,0 is determined in the Neutron_ Kinetics component by the reactor-kinetics equation (9), in which the effective, neutron-importance-weighted averages of the DNPs concentrations
equation (10)
are used, noting that, in the single-energy diffusion theory approximation, the neutron-importance function is taken as the neutron flux profile. P sinkPressure Flow1DFV_FuelSalt pumpFuel Qex X dnfiss drtot
b nfiss þ ¼ lg cg;eff þ S L dt g MSR Core Neutron Kinetics 5 R cg;eff ðtÞ ¼ Cold_leg Fig. 5. Object-oriented Modelica model of the fuel circuit. ∂w ∂p ∂z C f v þ A þ Adg þ wjwj ¼ 0 ∂x ∂x ∂x 2dA2 Ad ∂h ∂h 000 þ w ¼ Aðq000 fiss þ qDH Þ ∂t ∂x ∂cg w ∂cg bg þ ¼ nfiss c
lg cg ∂t Ad ∂x L g ¼ 1; . . . ; 8 ∂F k w ∂F k þ ¼ f k lDH;k q000 fiss
lDH;k F k Ad ∂x ∂t ð2Þ ð3Þ q000 fiss ðx; tÞ ¼ ð7Þ X F k ðx; tÞ: ð8Þ q000 DH ðx; tÞ ¼ k drtot ðtÞ ¼ drext ðtÞ þ drT ðtÞ þ drdens ðtÞ ð11Þ drT ðtÞ ¼ aT ½T eff ðtÞ
T eff;0  ð12Þ drdens ðtÞ ¼ adens ½T eff ðtÞ
T eff;0  ð13Þ k ¼ 1; 2; 3: ð5Þ Qfiss ðtÞ R cðxÞ Acore cdx ð10Þ The total reactivity
equation (11)
is the sum of the externally inserted reactivity drext and the feedback reactivity of fuel salt temperature and density. The latter two are determined by equations (12) and (13), where the effective temperature Teff is determined as a neutronimportance-weighted core average
equation (14)
and Teff,0 is the reference temperature with respect to which the reactivity defects are calculated. The effective delayed neutron fractions, which take into account the spatial distributions of the DNPs and the importance of the emitted neutrons, are evaluated according to equation (15). ð4Þ 000 The RHS source terms q000 fiss and qDH in equation (3) are the fission power density and the decay-heat generation density, respectively. The friction coefficient Cf appearing in the momentum equation (2) is evaluated using the Colebrook hydraulic correlation [21]. The term c is the neutron importance function and it was assumed to be fixed and equal to the fundamental eigenfunction of the single-energy diffusion theory for bare uniform reactor
i.e., a sinusoidal profile with a proper extrapolation length (Eq. (6)). The values of the fission 000 heat concentration q000 fiss and decay heat concentration qDH are computed from equations (7) and (8).
 x c ¼ cðxÞ ¼ sin p ð6Þ Le cðxÞcg ðx; tÞdx : R ½cðxÞ2 dx ð9Þ R T eff ðtÞ ¼ cðxÞT ðx; tÞdx R cðxÞdx ð14Þ R beff ðtÞ ¼ cðxÞlg cg ðx; tÞdx ) : X R nfiss ðtÞ lg cg ðx; tÞ dx cðxÞ L cðxÞ þ ( ð15Þ g The 16 external loops forming the fuel circuit were modelled as a single equivalent loop formed by a hot leg section, representing the piping from the core outlet to the IHX inlet, the IHX and a cold leg section representing the piping from the IHX outlet to the core inlet (Fig. 6). The HotLeg and Cold leg tube components implement the single-state, one-dimensional, finite-volume-discretized conservation equations for mass (Eq. (1)) and momentum (Eq. (2)), whereas energy (Eq. (16)), DNPs (Eq. (17)), and DH (Eq. (18)) equations are modified to consider only the decay term. Ad ∂h ∂h þ w ¼ vq00exch þ Aq000 DH ∂t ∂x ð16Þ 6 C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) especially the Flow1DFV component, which describes the fluid flow in a rigid tube. It is based on a 1D finite volume discretization of the mass (Eq. (20)), momentum (Eq. (21)) and energy transport (Eq. (22)) equations: A ∂d ∂w þ ¼0 ∂t ∂x ∂w ∂p ∂z C f v þ A þ Adg þ wjwj ¼ 0 ∂t ∂x ∂x 2dA2 Ad Fig. 6. Fuel circuit geometry. ∂cg w ∂cg þ ¼
lg cg ∂t Ad ∂x g ¼ 1; . . . ; 8 ∂F k w ∂F k þ ¼
lDH;k F k Ad ∂x ∂t k ¼ 1; 2; 3: ð17Þ ∂h ∂h þ w ¼ vq00exch : ∂t ∂x fuel circuit Since the fuel circuit forms a closed loop, it was important to provide an expansion tank to avoid strong pressure variations caused by temperature transients. The SinkPressure component allows handling any mass insurge or outsurge transient, with no associated dynamic effect. When mass flows from the sink to the loop, the outsurge fluid was assumed to be at the same temperature of the cold leg. 3.2 Intermediate heat-exchanger (IHX) Due to its non-conventional design, an effort was spent to set up a specific component representing the MSFR intermediate heat exchangers [10]. The heat exchangers were modelled as counterflow heat exchangers, with particular reference to the Printed Circuit Heat Exchanger
a proposed technology for the MSFR, for more detail see [11]
but any other counterflow arrangement based on parallel flow pipes subject to heat transfer through their lateral surface can be modelled as well with little modification. The Intermediate_HX model (Fig. 5) is based on components from the ThermoPower library, ð21Þ ð22Þ The geometrical parameters that can be specified in the component are the length L, the cross-section area A and the heat transfer perimeter v, which for a PCHE are expressed as A¼p ð18Þ Ideal, mass-flow-rate-controlled pumps (PumpFuel component) establish the salt flow through the circuit. The reactor total power is the sum of the fission power in the core and the decay heat generated along the whole fuel circuit
equation (19). Reactor geometrical, operational, physical and neutronic data used in the following are shown in Tables 1 and 2 (with reference to Fig. 6). All the parameters of the simulator are easily modifiable at runtime to allow for model modification and update throughout the various design phases. Z Aq000 ð19Þ Qreactor ðtÞ ¼ Qfiss ðtÞ þ DH ðx; tÞdx: ð20Þ d2ch 8 ð23Þ  p v ¼ dch 1 þ 2 ð24Þ where dch is the channel diameter. Figure 7 shows the Modelica model of the IHX whereas the geometric and operational parameters are shown in Table 3. Onedimensional finite-volume discretization with countercurrent flows was employed for the heat transfer in the heat exchanger. A single, equivalent heat exchanger component, representative of the 16 parallel ones (one for each of the parallel fuel circuit loops), was used. Longitudinal heat transfer along the flow direction was neglected, while the heat capacity of the metal walls of the heat exchanger was accounted for. Equations (25) and (26) are the heat exchange equations on the hot (fuel salt) and cold (intermediate salt) sides, respectively. Equation (27) is the energy balance equation for the IHX metal wall. q00hot ¼ hhot ðT fuel salt
T w;hot Þ ¼ k ðT w;hot
T vol Þ ð25Þ s=2    k  T w;cold
T vol ð26Þ q00cold ¼ hcold T int salt
T w;cold ¼ s=2   dT vol v q00hot þ q00cold ¼ Am dm cm : dt ð27Þ Due to the small channel size, the resulting flow is laminar in most of the cases for the fuel salt side. This simplifies considerably the heat transfer modelling (even if it restricts the heat transfer coefficients to quite low values). The average Fanning friction factor (Eq. (28)) and Nusselt number (Eq. (29)) for fully developed laminar flow in semicircular ducts [22] were implemented. In equations, it reads: 15:767 Re ð28Þ Nu ¼ 4:089: ð29Þ f¼ C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) 7 Table 1. Physical properties of fuel and intermediate salt [23,24]. Parameter Unit Value Fuel salt
LiF-ThF4- UF4-(TRU)F3 Melting point °C Density kg m
3 Dynamic viscosity Pa s Thermal conductivity W m
1 K
1 Specific heat capacity J kg
1 K
1 Intermediate salt
Fluoroborate (NaF-NaBF4/8%–92%) Melting point °C Density kg m
3 Dynamic viscosity Pa s Thermal conductivity W m
1 K
1 Specific heat capacity J kg
1 K
1 235 581 5108–0.8234 T (K) 6.187 · 10
4 exp (772.2/(T(K)–765.2)] 1.7 1010 694 2446.3–0.711 · T(K) 8.77  10
5exp(2240/T(K)) 0.66–2.37  10
4 · T(K) 1506 Table 2. MSFR geometric, operational and physical parameters. Parameter Symbol Value Geometric and operational parameters Core length LC 1.9 m Core radius RC 1.23 m Hot leg length LHL 0.65 m Hot leg radius RHL 0.15 m IHX length LHX 0.52 m Cold leg length
vertical LCLV 1.38 m Cold leg radius
vertical RCLV 0.15 m Cold leg length
horizontal LCLH 0.65 m Cold leg radius
horizontal RCLH 0.15 m Neutronic parameters DNP fraction
group 1 b1 12.3  10
5 DNP fraction
group 2 b2 71.4  10
5 DNP fraction
group 3 b3 36.0  10
5 DNP fraction
group 4 b4 79.4  10
5 DNP fraction
group 5 b5 147.4  10
5 DNP fraction
group 6 b6 51.5  10
5 DNP fraction
group 7 b7 46.6  10
5 DNP fraction
group 8 b8 15.1  10
5 DNP fraction
total b 459.7  10
5 Doppler feedback coefficient aT –1.46 pcm K
1 Decay heat parameters DH fraction
group 1 f1 0.0117 DH fraction
group 2 f2 0.0129 DH fraction
group 3 f3 0.0186 Parameter Symbol Value Reactor thermal power Fuel mass flow rate Intermediate mass flow rate Number of sectors Core inlet temperature Core outlet temperature Intermediate salt IHX inlet temp. Intermediate salt IHX outlet temp. Extrapolation Length Qreactor – – – Tcore_in Tcore_out Tint_min Tint_max 3000 MWth 29703 kg s
1 28458 kg s
1 16 675 °C 775 °C 670 °C 600 °C 0.10 m DNP decay constant
group DNP decay constant
group DNP decay constant
group DNP decay constant
group DNP decay constant
group DNP decay constant
group DNP decay constant
group DNP decay constant
group Neutron generation time Density feedback coefficient l1 l2 l3 l4 l5 l6 l7 l8 L adens 0.0125 s
1 0.0283 s
1 0.0425 s
1 0.133 s
1 0.292 s
1 0.666 s
1 1.63 s
1 3.55 s
1 6.65 × 10
7 s –2.91 pcm K
1 lDH,1 lDH,2 lDH,3 0.01973 s
1 0.0168 s
1 3.58  10
4 s
1 DH decay constant
group 1 DH decay constant
group 2 DH decay constant
group 3 On the cold (intermediate salt) side the flow regime is in the transition zone (Re ≈ 5000  7000), and the Gnielinski [21] correlation (Eq. (30)) is used. fDarcy(Re) is the Darcy friction factor, for which the Petukhov [21] correlation for smooth tubes (Eq. (31)) is used 1 2 3 4 5 6 7 8   f Darcy =8 ðRe
1000ÞPr Nu ¼  1=2   1 þ 12:7 f Darcy =8 Pr2=3
1 ð30Þ f Darcy ¼ ð0:79 lnðReÞ
1:64Þ
2 : ð31Þ 8 C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) 3.3 Intermediate loop and secondary heat-exchanger (SHX) Q_ex Fig. 7. Object-oriented Modelica model of the IHX. Table 3. Geometric and operational parameters of the IHX. Parameter Symbol Value Total heat transfer rate Intermediate circuit higher temp. Intermediate circuit lower temp. Width Height Length Channel diameter Plate thickness Channel pitch Channels number Global heat transfer coeff. Hot channel Reynolds number Cold channel Reynolds number Hot channel average velocity Cold channel average velocity Hot channel pressure drop Cold channel pressure drop Fuel salt volume – Tc,out Tc,in W H L dch tp pch nch UA Reh Rec uh uc Dph Dpc Vsalt 187.5 MW 670 °C 600 °C 1m 1.5 m 0.69 m 2 mm 1.3 mm 2.5 mm 240,000 3.02 MW K
1 238 5631 1.14 m s
1 2.66 m s
1 3.96 bar 1.30 bar 0.26 m3 The thermohydraulic correlations to be used in the IHX component are selectable at runtime, to allow for design variations in geometrical and/or operational IHX parameters. The four intermediate loops were modelled as a single equivalent loop formed by a hot leg section, representing the piping from the IHX outlet to the SHX inlet, a bypass line and a cold leg section representing the piping from the SHX outlet to the IHX inlet (Fig. 8). The intermediate loop model was assembled by using standard components from the ThermoPower library. The adopted scheme is represented in Figure 8. The two basic components are the hotLeg and coldLeg components, which are modelled by Flow1DFV objects. The transport delay associated with the hot/cold leg, with the geometrical parameters indicated in Table 4, is of the order of some seconds. In addition, the dynamic effect associated with thermal inertia is not negligible, hence the total volume of the intermediate loop has a significant influence on dynamics. The loop includes two active components, a pump and a bypass valve (Fig. 8). The pump class models a simple centrifugal pump with no energy or momentum dynamics and the power consumption was simply estimated through a constant pump efficiency hp. The pump has an external input port which can be used to control the rotational speed and thus the mass flow rate. The valve component was modelled by the ValveLin class, which simply provides a linear constitutive equation to relate the pressure drop Dpv and the bypass mass flow rate Gbypass: Gbypass ¼ K v cmd Dpv ð32Þ where Kv is a hydraulic conductance parameter set to 10
2 and cmd is the command signal, provided by an external input port. The valve can be used to control the mass flow rate flowing in the secondary heat exchanger, providing another way to control heat extracted from the intermediate loop. As explained in Section 3.1 for the fuel circuit, an expansion tank was provided to avoid the strong pressure variations related to temperature transients of an incompressible liquid in closed loop and to establish a reference pressure level in the cold leg (1 bar). The expansion tank was modelled using the expansionTank component (Fig. 8). The other components appearing in Figure 8 are simple temperature and mass flow rate sensors, which model zeroorder sensors providing ideal measurements. Geometric and operational parameters of the intermediate loop are shown in Table 4. In the SHX, heat is transferred from the intermediate salt to the helium in the Energy Conversion System (ECS). The modelling approach employed for the SHX was identical to that used for the IHX (see Sect. 3.2). Geometric and operational parameters are shown in Table 4. The flow regime in the SHX is fully turbulent on both the salt and gas sides (ReD ≈ 4  104 and ReD ≈ 1.5  105, respectively). The Gnielinski [21] correlation is used to evaluate the convective heat transfer coefficients. Also in this case, the correlations to be used in the SHX component are selectable at runtime, to allow for design variations in geometrical and/or operational SHX parameters. C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) 9 Fig. 8. Object-oriented Modelica model of the intermediate loop. 3.4 Energy conversion system (ECS) 3.4.1 Compressor The energy conversion system model was assembled by using standard components from the ThermoPower library. In particular, a Helium Joule-Brayton cycle with regeneration and three stages of reheating and intercooling was considered. This configuration turned out to ensure a gas temperature at secondary heat exchanger inlet that can avoid salt solidification problem [11]. The adopted scheme is represented in Figure 9. There are five main components in the model, namely, the compressor, the turbine, the intercooler, the reheater, and the recuperator (Fig. 9). The cycle was modelled as open, i.e., disregarding the final heat sink section. This is a common choice to simplify the modelling of the cycle [5] and it has no impact on the dynamics results since the final sink acts as an infinite heat sink. The compressor was modelled by considering an energy balance. Since Helium can be considered as perfect gas, the following relations hold: T out;c ¼ T in;c þ T iso;c  1 T iso;c
T in;c hc
g
1 pout;c g ¼ T in;c pin;c ð33Þ ð34Þ where Tin,c and pin,c are the gas temperature and pressure at the inlet of the compressor, Tout,c and pout,c are the gas temperature and pressure at the outlet of the compressor, hc is the compressor efficiency, Tiso,c is the isentropic outlet temperature of the compressor and g is the specific heat ratio of the gas. The efficiency and the pressure ratio can be 10 C. Tripodo et al.: EPJ Nuclear Sci. Technol. 5, 13 (2019) Table 4. Geometric and operational parameters of the SHX. Parameter Value Intermediate loop Hot leg length Hot leg radius Cold leg length
horizontal Cold leg radius
horizontal Cold leg length
vertical Cold leg radius
vertical Secondary heat exchanger Total heat transfer rate Intermediate circuit higher temp. Intermediate circuit lower temp. Inlet gas temperature Outlet gas temperature Length Channel diameter Plate thickness Channels number 5m 0.75 5m 0.75 0.82 0.75 set by the user in order to adapt the component to the cycle parameters. The component can be connected to a shaft in order to calculate the compressor work (and hence the cycle efficiency). 3.4.2 Turbine m m m m 750 MW 670 °C 600 °C 460 °C 615 °C 1.51 m 10 mm 6 mm 250,000 The turbine was modelled by considering an energy balance similar to that used in compressor component. In particular, T out;t ¼ T in;t
ht ðT in;t
T iso;t Þ
T iso;t ¼ T in;t pout;t pin;t g
1 g ð35Þ ð36Þ where Tin,t and pin,t are the gas temperature and pressure at the inlet of the turbine, Tout,t and pout,t are the gas temperature and pressure at the outlet of the turbine, ht is the turbine efficiency, Tiso,t is the isentropic outlet temperature of the turbine. Also, in this case, the efficiency and the pressure ratio are user-selectable parameters and the turbine work can be calculated. mass_flow_reheat Reheater_1 RH Reheater_2 RH Reheater_3 RH Speed Power 314.15926535898 power turbine_1 turbine_2 turbine_3 turbine_4 Recuperator REG P compressor_4 compressor_3 compressor_2 compressor_1 sink He_mass_flow_rate sourceMassFlow Intercooler_3 Intercooler_2 Intercooler_1 T_intercooling Fig. 9. Object-oriented Modelica model of the ECS.