*Nuclear Reactors - Spacecraft Propulsion, Research Reactors, and Reactor Analysis Topics*

tables, or customized material implemented in MOOSE source code. The BISON fuel performance code provides a variety of material models for nuclear materials [61]. BISON offers material properties for W and Mo-30W CERMETs [62].

The heat conduction module provides different interfaces for representing conjugate heat transfer. It can be applied as a boundary condition on channel boundaries or it can be lumped into a volumetric term. The coupling with the thermalfluids code RELAP-7 [63] can be performed using a Robin-Neumann boundary or a Robin-Robin boundary strategy.

### *4.2.3 Thermal mechanics*

Stresses in NTP systems arise from large temperature gradients, mechanical contact during transient and steady-state operation, and pressure differential over the core. The mechanical problem is a coupled problem between heat conduction, mechanics, contact, and potentially thermal-fluids. Vibrations can manifest in the solid structures that interact with fluid pressure oscillations caused by turbomachinery, flow separation, or other fluid-mechanical effects. The material properties relevant in mechanical problems include Young's modulus, Poisson's ratio, the linear expansion coefficient, and parameters describing plastic deformation, such as the yield stress and hardening law; these material properties generally depend on temperature.

MOOSE provides the capability to conduct mechanics simulations in the *tensor mechanics* module [64]. The tensor mechanics module is seamlessly able to couple with the heat conduction module, facilitating thermal-mechanics simulations. MOOSE also implements a variety of mechanical contact algorithms in its *contact* module. Finally, MOOSE allows pluggable multiphysics capabilities coupling neutronics, heat conduction, and time-dependent mechanics [65].

### *4.2.4 Thermal fluids and balance of plant*

Nuclear thermal propulsion in its current form in the U.S. uses a HALEU-based reactor core to generate several hundred megajoules of thermal energy to heat hydrogen propellant to high exhaust temperatures for engine thrust. NERVA designs up to current engine concepts are of an *expander cycle* design; **Figure 4** shows a simplified representation of an NTP expander cycle engine.

In this design, high pressure liquid hydrogen (H2) is pumped from storage tanks and is preheated while used to cool the nozzle, reactor pressure vessel, reflector and control drums and control drums (converting it to gaseous H2), using the energy added to the gas to drive turbines. The exhaust from the turbine is directed to core support and shielding structures (not shown in **Figure 4**). Next, the gas passes through the coolant channels in the individual coolant block comprising the reactor core, where it is superheated to the necessary high exhaust temperatures. Finally, the gas is expanded through a nozzle with a high nozzle area ratio to generate thrust. Thrust is maximized by maximizing the gas temperature exiting the core, but current reactor material performance limits will restrict the peak temperature to something less than about 3000 K [44].

Unlike power reactors, NTP engines are expected to operate continuously for less than an hour at a time with weeks to months between burns [66]. Each operational period will consist of three phases: startup to full power, full thrust operation, and shutdown (with decay heat removal). Flow rates are matched to the reactor power according to the demands of each period. During startup, hydrogen economy requires as rapid an ascent to full power as possible through appropriate control drum rotation, and H2 flow is used to both cool the reactor, as well as protect other

*Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.103895*

**Figure 4.**

*Representation of NTP engine system with (0) liquid hydrogen storage tank, (1) pre-heated-hydrogen-driven turbopump, (3) nozzle cooling, (4) pressure vessel/reflector/control drum cooling, (5) gaseous hydrogen feed to turbopump, (6) gas plenum above core, (7) reactor core and hydrogen cooling, and (8) exhaust nozzle.*

engine components. During the full thrust period, the core and balance of plant are near steady-state conditions. At shutdown, the reactor will be returned to a subcritical state, but hydrogen flow will be needed for decay heat removal.

The M&S capabilities required for the thermal-fluids and balance of plant are: ability to exchange heat with solid conduction (i.e., conjugate heat transfer), modeling hydrogen in a temperature range from 40 to >3000 (or greater than 3000) K, ability to model compressible flow, availability or extendability to include heat transfer and pressure drop correlations suitable for NTPs, ability to model the relevant components in the NTP system (e.g., turbo pump and turbine on common shaft, valves, etc.), and a flexible control system that allows for the simulation of complex controllers.

RELAP-7 can solve single-phase (e.g., 3-equation model) and two-phase (e.g., 7 equation model) system analysis problems using a discontinuous Galerkin HLLC (Harten, Lax, and Van Leer Contact) discretization [67]. RELAP-7 provides models for a variety of components, including pipes, pumps, valves, and turbines; in addition, it supports both full (i.e., single nonlinear problem) and tight (i.e., Picard type) coupling with MOOSE heat conduction solvers via conjugate heat transfer. RELAP-7 provides *para-hydrogen* fluid properties across the required range, and provides a flexible and extendable control system that can be used to simulate the control system for an NTP model.

#### **4.3 Case study of a reactor startup simulation with MOOSE**

In this section, Griffin, RELAP-7, and MOOSE modules are coupled and used for a simulated startup of a LEU, CERMET-based core similar to the one depicted in **Figure 5**, but with an operating power of 250 MW and an approximate thrust of 55,600 N (12,500 lbf). The core consists of 61 LEU fuel assemblies arranged in five circular rings within a zirconium hydride (ZrH) monolithic moderator block. The startup simulation includes a PID-controlled rotation of the drums to match a predetermined reactivity setpoint curve, neutronics modeled with diffusion and Super-Homogenization (SPH) [68], heat conduction, and thermal-fluids.

From a neutronics standpoint, the probabilities of neutron interaction represented by cross-sections are affected by several temperature-driven feedback mechanisms. For the reactor shown in **Figure 5**, the primary feedback comes from the increase in 238U capture reactions as the fuel heats up (Doppler feedback) while other important feedback mechanisms are spectral shift and hydrogen content in the core. From a modeling perspective, spectral shift and changes in moderator content are more difficult, because their effect is global. The value of the

*Nuclear Reactors - Spacecraft Propulsion, Research Reactors, and Reactor Analysis Topics*

**Figure 5.** *Concept of BWXT NTP reactor design (picture courtesy [34]).*

**Figure 6.** *Full-core serpent model. (a) Geometry; (b) fission rate and thermal flux.*

## *Nuclear Thermal Propulsion DOI: http://dx.doi.org/10.5772/intechopen.103895*

temperature or hydrogen density at one point affects the neutron spectrum, and thus, the effective cross sections at another point. The distance over which nonlocal effects materialize depend on how far a neutron can travel without being absorbed or escape the reactor. For this kind of reactor, this travel distance, or mean free path, can be quite large (on the order of 1–100 cm, depending on the neutron energy) and complicates the cross-section evaluation significantly, especially as the tremendous axial thermal gradient gives perceptibly different neutron spectra in different parts of the core. Therefore, the analyst may opt to tabulate cross sections not only for different temperatures and hydrogen, but also for different shapes of the temperatures and hydrogen densities. For this example, cross sections are pre-tabulated for different values of the important feedback variables (e.g., fuel, moderator and reflector temperatures, control drum angle). The Serpent Monte Carlo code is used for tabulating the cross-sections for this work [69] Plots from the Serpent model are shown in **Figure 6**.

The accuracy of the solution and execution time of the model are balanced by representing the neutron distribution by the neutron diffusion equation, discretizing it on a coarse mesh, and using the full-core SPH in Griffin. SPH can be seen as a physics-based reduced order modeling approach. This enables the use of a coarse numerical mesh, as shown in **Figure 7**, while preserving the key quantities of interest needed for the multiphysics coupling, such as reactivity and power density distribution.

The moderator monolith is not expected to see a large temperature increase compared with the fuel because each of the fuel assembly is surrounded by a layer of insulator. For preliminary calculations, it is thus acceptable to assume that fuel assemblies exchange little heat with one another. Due to various symmetries, the conductive and radiative heat transfer over each ring of fuel assemblies is therefore simulated by a single 30° slice, shown in **Figure 8** and extruded over the entire height of the active core. In this figure, the orange, red, green and blue regions correspond to the fuel, insulator (ZrC), shell (SiC), and moderator (ZrH), respectively. The fuel region is penetrated by 127 cooling channels. The moderator is also cooled by flow channels to remove most of the heat that radiatively crosses the three gaps between the fuel and the moderator. The thermal-fluids is modeled by two representative cooling channels per fuel assembly ring to simulate the convective heat removal in the fuel and in the moderator.

The integration of the various sub-modules into a multiphysics model is summarized in **Figure 9**. The neutronics model provides the power density into each of the 30° slice thermal models (e.g., one per ring). These provide the wall temperature to their respective cooling channels, which in turn provide the fluid temperature and heat transfer coefficient needed to evaluate the amount of heat removed by the coolant. Once the thermal field in each of the representative fuel assemblies is obtained, the fuel and moderator temperatures are passed back to the neutronics model to update the cross sections accordingly.

To perform a reactor start-up, the control drums need to be rotated to add sufficient reactivity to not only increase the reactor power, but also compensate for the negative feedback ensuing from the heat-up of the fuel. Attempting to select the rotation of the drums *a priori* to obtain a desired power evolution would likely require significant trial and error iterations, especially considering the nonlinear behavior of the reactivity feedback coefficients and fuel heat capacity as a function of temperature. Rather, efficient control of the drums can be achieved through automated means—for instance relying on a widely-used Proportional-Integral-Derivative (PID) controller, as illustrated in **Figure 10**. Given a desired power setpoint, it can be converted into a reactivity signal (~*ρ* in **Figure 10**), which is then compared to the *measured* reactivity from the model. This measurement

**Figure 7.** *Full-core neutronics and thermal meshes.*

**Figure 8.** *X-Y view of the 30° slice thermal mesh.*

**Figure 9.**

*Schematics of the full-core multiphysics model.*

**Figure 10.** *Schematics of the PID control of the full-core multiphysics model.*

corresponds to the reactivity computed by the numerical model with, for instance, an additional typical time delay from the detectors. An error between the desired and measured reactivity is then computed. The updated control drum angle is determined by adding three terms proportional to: (1) the error to attempt instantaneous correction; (2) the integral of the error to account for any persistent underestimating/overestimating of the desired reactivity; and (3) the derivative of the error to anticipate how it is going to evolve in the near-future and avoid overcorrection, with the controlling constants called *Kp*, *Ki*, and *Kd*, respectively.

The reason the reactivity is chosen to control the PID—rather than the power is that a rotation of the drums induces an immediate reactivity change, whereas the corresponding power response is quite delayed (e.g., one may consider reactivity as being roughly the derivative of the power with respect to time). As such, it results in a much more stable control system. However, measured and desired power can be relatively easily converted to reactivity if the neutronic kinetics parameters of the reactor are well known.

The optimal values of *Kp*, *Ki*, and *Kd* can theoretically be determined if the transfer function for the system is known. However, given the complexity of the multiphysics model, it appears impractical to proceed that way. Instead, their values are chosen based on a semi-empirical approach. In particular, *Kp* represents the angle by which the drums are to be rotated per amount of reactivity. Fortunately, in most of the realistic operational range of the drums, the reactivity inserted per degree (*α*) is fairly constant and *Kp* can be approximately set to 1*=α*. If the error consistently lags behind the set point or tends to over-correct, the proper approach is to adjust *Ki* or *Kd*. In any event, the values of *Kp*, *Ki*, and *Kd* can be adjusted to make the system more or less responsive.

#### **Figure 11.**

*Reactivity setpoint, actual system reactivity, control drum actuation, power response, fuel average temperature, outlet coolant temperature, and moderator average temperature of the generic CERMET NTP system during the startup transient. (a) Reactivity control; (b) Core heating.*

In the current simulations, the control system compensates the change in feedback accompanying the change in power well. During the simulation of a startup transient, the reactivity set point is chosen to be 0.3\$ for the first 50 s, linearly ramping down to 0.2\$ by 80 s of startup, and then remaining constant afterwards. The actual reactivity observed in the simulation closely follows the reactivity set point until the maximum control drum rotation is reached at about 100 s.

Reactor power increases from the initial 610 kW (10 kW per assembly) to close to 250 MW without over-swings in the completed simulation time. At around 90 s, a local maximum in the power is assumed that is attributed to the negative feedback outrunning control drum motion compensating for it. In this case, reactivity is under-compensated.

Temperatures increase monotonically throughout the transient with a corresponding temperature rise in the fuel, and outlet hydrogen being the largest at about 1500 K and moderator temperature rise being very small at less than 120 K. Increase in power will likely have to occur quicker in some NTP operational scenarios. It remains to be investigated if temperatures remain monotonic in these scenarios. The Griffin/RELAP-7/MOOSE model described herein is well equipped to investigate these scenarios (**Figure 11**).
