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Small modular reactors (SMRs) are those fission reactors whose electrical output power is no more than 300 MW_{e}. SMRs usually have the inherent safety feature that can be applicable to power plants of any desired power rating by applying the multimodular operation scheme. Due to its strong inherent safety feature, the modular high temperature gas-cooled reactor (MHTGR), which uses helium as coolant and graphite as moderator and structural material, is a typical SMR for building the next generation of nuclear plants (NGNPs). The once-through steam generator (OTSG) is the basis of realizing the multimodular scheme, and modeling of the OTSG is meaningful to study the dynamic behavior of the multimodular plants and to design the operation and control strategy. In this paper, based upon the conservation laws of mass, energy, and momentum, a new differential-algebraic model for the OTSGs of the MHTGR-based multimodular nuclear plants is given. This newly-built model can describe the dynamic behavior of the OTSG in both the cases of providing superheated steam and generating saturated steam. Numerical simulation results show the feasibility and satisfactory performance of this model. Moreover, this model has been applied to develop the real-time simulation software for the operation and regulation features of the world first underconstructed MHTGR-based commercial nuclear plant—HTR-PM.

Nuclear fission energy is a crucial type of clean energy sources that can meet the world’s increasing energy demands and also address challenges associated with global climate and environmental impact. After the successful development of small (tens of megawatts) light water reactors (LWRs) for propulsion by the U.S. Navy, the commercial fission reactors began to commission in the late 1950s and early 1960s, which were essentially the scaled-up versions of those naval nuclear reactors [_{e}. Due to the low power density and large heat capacity, SMRs usually have the inherent safety feature, and could be beneficial in providing electricity power to remote areas without transmission or distribution infrastructure, in generating local power for a large population center and in being viable for specific applications such as heat sources for the industrial complexes [

There are three major groups of SMR designs that are actively being developed in the US, China, Japan, Korea, and other countries. As shown in Figure _{th} pebble-bed high temperature gas-cooled reactor HTR-10, which was developed by the institute of nuclear and new energy technology (INET) of Tsinghua University, achieved its full power-level in 2003 [

Schematic views of some LWR-type SMRs.

IRIS

NuScale

mPower

SMART

4S nuclear energy system.

Reactor

Plant structure

OTSG

Structure of the HTR-PM plant.

From the above introduction, it is clear to see that the OTSG is key equipment in the SMR-based nuclear energy systems. Actually, this is given by the necessity of building large nuclear plants based on parallel-operated multi-SMRs. The precondition of applying this parallel operation scheme is that the pressure of the steam generated by each NSSS must be equal to each other. Since the widely-utilized U-tube steam generators (UTSG) can only provide saturated steam whose pressure and temperature must obey the one-to-one relationship, parallel-operating SMRs based on the UTSGs may lead to the drift in the steady values of the coolant temperatures [

The above OTSG models are all developed under the assumption that the outlet steam is superheated. However, in some actual cases such as the plant startup or operation in very low power, the OTSG might also generate saturated steam. The OTSG dynamics in these cases is quite significant for designing the plant operation strategy, but it cannot be described by the models presented in [

The OTSG of MHTGR-based multimodular nuclear plants such as the HTR-PM is a helical-coil once-through shell-and-tube counterflow heat exchanger. The hot helium flows into the primary shell side, and the cold water is fed into the secondary tube side. The hot helium transfers the thermal power from the primary side to the secondary water/steam two phase flow, which results in cooling the helium and heating the feedwater to superheated or saturated steam. Here, it is assumed that the two phase flows in all the tubes of the OTSG are identical, which means that the OTSG can be simply treated as a single tube heat exchanger. In order to obtain dynamic model that is suitable to not only the case of providing superheated steam but also the case of generating saturated steam, the OTSG is divided into two sections, that is, the subcooled section and boiling section. This nodalization scheme is illustrated in Figure

Nodalization scheme of the OTSG.

Then, under the coordinate frame given in Figure ^{3}), ^{2})), ^{2}), and ^{2}). The OTSG dynamic model in this paper is obtained based upon the conservation laws determined by (

Moreover, to obtain the lumped parameter model of the OTSG based on integrating (

For a continuously differentiable function

Based upon conservation laws (

The dynamic equations of both the subcooled and boiling sections of the OTSG secondary side are established in this subsection.

In the subcooled section, integrate (

Moreover, since the dynamic behavior of the pressure-flow process is much faster than that of the enthalpy-temperature process, the time-differentiation term in (

Then, by integrating (

Since the water near point 3 in Figure

Moreover, it is not loss generality to assume that

Based on (

By considering the transport inertia in the subcooled section, the dynamics of the secondary-side flowrate and water enthalpy at point 1 can be described by

Due to the high velocity and relative low density of the fluid in the boiling section, we assume that

From Lemma

Based upon relationship between (

Define the state-vector and input-vector of the OTSG secondary side as

Then, based upon (

Since there is no flow inside the metal tube wall between the primary and secondary sides, it is clear that partial differential equation (

Here, for the simplicity of the model, we assume that the tube temperature is linearly distributed along

From (

By defining

According to the nodalization scheme illustrated in Figure

From the conservation law of energy, it is clear that

Moreover, from the energy balance relationship between the primary and secondary side, it is clear to see that

Moreover, assume that the helium temperature distribution along the

Based on (

Based upon the above analysis and derivation, define the state-vector

In the process of developing dynamic model (

In order to verify the feasibility of differential-algebraic OTSG model (

The main steady-state results at the operating power-levels of 100%, 90%, 75%, 50%, and 30% reactor full power (RFP) with comparison to the designed values are given in Table

Comparison of the steady simulation results and designed data.

Power-level/RFP | 100% | 90% | 75% | 50% | 30% |
---|---|---|---|---|---|

Helium flowrate |
96.26 | 83.95 | 72.58 | 51.3 | 33.44 |

Feedwater flowrate |
93.25 | 84.24 | 70.75 | 48.76 | 30.93 |

Hot helium temperature |
750.0 | 743.5 | 733.75 | 717.5 | 704.5 |

Feedwater temperature |
205.2 | 199.1 | 191.1 | 175.1 | 157.1 |

Designed outlet steam temperature |
571.0 | 571.0 | 571.0 | 571.0 | 571.0 |

Simulated outlet steam temperature |
571.0 | 571.0 | 571.0 | 571.0 | 571.0 |

Relative error of steam temperature/% | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Designed cold helium temperature |
243.0 | 237.2 | 228.5 | 214.0 | 202.4 |

Simulated cold helium temperature |
242.5 | 236.5 | 229.4 | 215.2 | 200.0 |

Relative error of cold helium temperature/% | 0.20 | 0.30 | 0.39 | 0.56 | 1.19 |

For verifying the feasibility of model (

Simulation results in case A.

Simulation results in case B.

Simulation results in case C.

By comparing the simulated and designed values given in Table

In case A, since the inlet hot helium temperature is constant, the step decrease in the helium flowrate results in the decrease of the heat transferred from the primary to the secondary side, which then induce the decreases in the outlet steam temperature, steam quality and length of the boiling section. Moreover, thermal power transferred from the primary to the secondary sides is also reduced, which further causes the decrease of the primary average helium temperature. Since the inlet hot temperature is not changed, the decrease in average helium temperature certainly results in the decrease of outlet cold helium temperature. From Figure

In the case of feedwater flowrate decrease at 75% RFP, since both the inlet helium temperature and primary helium flowrate remain constant when the step decrease in the feedwater flowrate occurs, the thermal power transferred from the primary helium flow to the tube-wall is not changed at the beginning. Then, the decrease in the feedwater flowrate must lead to the increases of the outlet steam temperature and steam quality and lengthening of the boiling section. Since the specific enthalpy of the feedwater is constant, the increase of the steam temperature reduces the temperature difference between the two sides of the OTSG, which leads to the decrease of the thermal power transferred from the primary to the secondary sides, and certainly further results in the increase of the temperature of the primary outlet helium flow. From Figure

In case C, after the occurrence of the large step increase in the feedwater flowrate, the outlet steam temperature is quickly and largely decreased since the thermal power transferred from the primary side is nearly not changed at the initial stage. The decrease in the steam temperature is certainly equivalent to the decreases of the boiling section length and steam quality, and results in a larger temperature difference between the two sides of the OTSG. This difference certainly enlarges thermal power transferred to the secondary side, and then results in the temperature decrease of the primary outlet cold helium. The above analysis well accords with the numerical results in Figure

Finally, from the above discussion, we can easily see that the steady precision of dynamic model (

Real-time simulation software for the HTR-PM operation by using model (

Due to the inherent safety feature of the SMR, SMR-based nuclear plants are an important developing trend of the nuclear energy systems. Based upon the multimodular operation strategy, SMRs can be used to build nuclear plants with any desired power rating and inherent safety. The OTSG is key equipment of any SMR-based multimodular nuclear plants, and developing the dynamic model for the OTSG is very meaningful to study the dynamic behavior of the multimodular nuclear plants. In this paper, based on the conservation laws of mass, energy, and momentum, a differential-algebraic model for the OTSG of those MHTGR-based multimodular nuclear plants is presented. This model can describe the dynamic behavior of the OTSG in both the cases of providing superheated steam and generating saturated steam. Moreover, since the relative simplicity of this model, it can also be applied to design steam temperature control laws for the OTSGs. Numerical simulations results show that this model has a satisfactory steady precision, and its transient responses well accord with the thermodynamic behavior of the OTSG. This model has already been applied to develop a real-time simulation software for the operation and control strategy design and verification of the HTR-PM plant. In the future, more simulation results of the closed-loop dynamic responses will be given by coupling this new OTSG model with steam temperature control laws.

The author declares that there is no conflict of interests regarding the publication of this paper.

_{th}HTR-PM demonstration plant