Manipulate core radius, reflector thickness, diffusion length, and k-infinity to visualize neutron flux distribution and calculate effective multiplication factor keff.
Manipulate core radius, reflector thickness, diffusion length, and k-infinity to visualize neutron flux distribution and calculate effective multiplication factor keff.
The core behavior is governed by the one-group neutron diffusion equation. It balances neutron production from fission against losses from absorption and leakage.
$$D_c \nabla^2 \phi_c - \Sigma_{a,c}\phi_c + k_{\infty}\Sigma_{a,c}\phi_c = 0$$Where for the core region: $D_c$ is the diffusion coefficient, $\Sigma_{a,c}$ is the macroscopic absorption cross-section, $\phi_c$ is the neutron flux, and $k_{\infty}$ is the infinite multiplication factor (a key material property you adjust in the simulator). The term $D_c \nabla^2 \phi_c$ represents neutron leakage.
In the reflector, there is no fission source, so the equation simplifies. The solutions in both regions must match at the boundary.
$$D_r \nabla^2 \phi_r - \Sigma_{a,r}\phi_r = 0$$Here, $D_r$ and $\Sigma_{a,r}$ are for the reflector. The critical condition—solving these equations with the correct boundary conditions—yields the value for $k_{eff}$. This is the eigenvalue the simulator calculates in real-time as you change parameters.
Reactor Core Design: Engineers use this exact one-group diffusion theory for preliminary sizing of new reactor cores. By adjusting core radius and material properties, they can estimate the minimum amount of fuel needed to achieve criticality, saving millions in prototyping costs.
Safety Analysis: Understanding flux distribution is crucial for predicting "hot spots" where power and temperature could be highest. A peaked flux profile might require different cooling strategies than a flat one, influencing the placement of coolant channels and safety systems.
Reflector Optimization: In small modular reactors (SMRs) and naval reactors, space is at a premium. Analyzing how reflector thickness impacts core size and $k_{eff}$ helps designers create compact, efficient power units without compromising safety.
Fuel Cycle Management: As fuel burns up in a reactor, $k_{\infty}$ of the core material decreases. Operators use diffusion models to predict how control rods need to be adjusted over time to compensate and maintain $k_{eff} = 1.0$, ensuring steady power until the next refueling.
First, keep in mind that the "one-group" model is not a panacea. It is an ultra-simplified model dealing with an "average neutron." For instance, fast neutrons and thermal neutrons diffuse and are absorbed in completely different ways, yet they are lumped together here. Therefore, it is optimal for qualitative trends and sensitivity analysis but cannot be used for detailed design of an actual reactor. As a next step, you need to use a multi-group diffusion code.
Next, a common pitfall in parameter setting is confusing the "Diffusion Length L" with the "Migration Area M²". The simulator has sliders for both, but L² is the mean square distance a neutron travels "from the end of slowing-down until its absorption." On the other hand, M² is the mean square distance "from birth in fission to absorption (including the slowing-down process)." In many cases, the relation M² ≒ L² + τ holds (where τ is the Fermi age). For example, in a light water reactor, L is on the order of a few cm, while M is on the order of tens of cm. If you are not aware of this difference, you will misinterpret the physical meaning of changing these parameters.
Finally, beware of the misconception that "if you just increase k∞ enough, anything will go critical." While k∞ indeed represents material properties, if the reactor size is too small, leakage becomes dominant, and you might not be able to bring keff to 1 no matter how much you increase k∞. Conversely, once you exceed a certain size (the critical size), keff asymptotically approaches k∞. Experiencing this tug-of-war between "leakage" and "material" is a key strength of this tool.
The "diffusion equation," which forms the core of this simulator, actually appears in many transport phenomena beyond reactor physics. For example, consider impurity diffusion in semiconductor processing. The process of doping a silicon wafer and then allowing the dopants to diffuse via heat treatment is precisely described by the diffusion equation. While there are differences, such as the absence of a source term ($\nabla^2 C = (1/D) \partial C/\partial t$), mathematically they are siblings.
Furthermore, the calculation to find the bell-shaped neutron flux profile is mathematically very similar to eigenvalue analysis in structural mechanics. Rearranging the diffusion equation yields the Helmholtz equation, $\nabla^2\Phi + B_g^2\Phi = 0$, which is identical in form to the equation for finding the vibration modes (natural frequencies and mode shapes) of a fixed-edge membrane. The reactor's "criticality" corresponds to the equation's "eigenvalue," and the neutron flux distribution corresponds to the "eigenfunction." If you are a CAE engineer, you should see how this connects to the essence of the problems your FEM solver solves.
Broadening the applications further, similar equations are used in photon diffusion theory (biomedical optical imaging) and modeling groundwater flow and contaminant transport. In other words, what you learn with this tool is the first step towards understanding the universal physical concept of "balance between generation, diffusion, and removal."
The logical next step is to study "two-group diffusion theory." By separating neutrons into a fast group and a thermal group and explicitly including the "slowing-down" process between them, you can start to understand crucial real-world reactor concepts like "spectral effects" and the role of the moderator. After playing with NovaSolver, challenge yourself with questions like: "Which is larger, the diffusion length of the fast group or the thermal group?" and "Why is fuel arranged in a lattice pattern in light water reactors?"
If you want to deepen the mathematical background, I encourage you to thoroughly review "separation of variables" and "handling boundary conditions." The spherical geometry equation solved by this simulator is tackled by separation of variables, setting $\Phi(r) = R(r)$, and solving the ordinary differential equation for $R(r)$. Its solution takes the form $\sin(B_g r)/(B_g r)$, and the boundary condition $\Phi(a)=0$ yields $B_g = \pi/a$. If you can follow this mathematical flow yourself, you will gain insight into applying these concepts to other geometries like cylindrical cores.
Ultimately, I recommend trying to read the input decks of codes used in practice (e.g., MONJU, SRAC, MVP). You will find definitions for dozens to hundreds of energy groups, detailed fuel compositions, and complex core layouts. This will make you keenly aware of how much the "one-group," "homogeneous core" assumptions of NovaSolver simplify reality. At the same time, precisely because you have learned the "essence of that simplification," you will be able to understand the meaning of each piece of data in those complex inputs.