Control Rod Worth Curve (2D)

Rod Bank Control

Insertion depth 0%

Bank worth (fully inserted) 900 pcm

Drive speed 6 %/s

Core State

Insertion depth0.0 cm

Differential worth dρ/dx0 pcm/%

Integral worth inserted0 pcm

k_eff1.00000

How it works

A control rod is a localized neutron absorber. First-order perturbation theory says the reactivity worth of a thin slice of rod at height z is proportional to the square of the unrodded neutron flux at that height, φ(z):

dρ/dx ∝ φ(z)²,   φ(z) = sin(πz/H)   (chopped-cosine axial shape)

Differential worth: dρ/dx(x) = -W · (2/H)·sin²(πx/H)
Integral worth:      ρ(x)    = -W · [ x/H - sin(2πx/H)/(2π) ]

Because flux peaks at the core mid-plane, a rod tip moving through the center of the core sweeps through the highest-flux region and adds reactivity fastest — the bell-shaped differential curve. Integrating it gives the classic S-shaped integral worth curve: shallow near both ends of travel, steepest through the middle third. This is exactly the curve reactor engineers measure during rod-worth calibration on a real plant.

Insertion depth — how far the bank has moved down from fully withdrawn (0%) toward fully inserted (100%).
Bank worth — the total reactivity (in pcm, 10⁻⁵ Δk/k) the bank is worth once fully inserted; real PWR banks range roughly 300–2000 pcm depending on design.
k_eff is computed from ρ = (k−1)/k, taking the all-rods-out state as the k = 1 reference.
The core strip's shading traces local axial neutron flux; the rodded region and the depression just below the rod tip are darkened, matching real in-core flux-mapping traces.

High flux Low flux / rodded Control rod