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Nuclear Engineering

Nuclear engineering formulas and calculators for reactor physics, radiation, and nuclear decay used in reactor design and safety analysis. Includes half-life, binding energy, and neutron flux equations.

96Formulas
8Subcategories
Updated Jul 2026
All (96) Fusion Engineering (9) Health Physics & Dosimetry (9) Nuclear Decay (9) Nuclear Fuel Cycle (9) Nuclear Thermodynamics (6) Radiation (11) Radiation Protection (10) Reactor Physics (15)

Health Physics & Dosimetry

External Radiation Dose Calculator

D = D˙ × t × (1 - shielding factor)

Health Physics & Dosimetry

Internal Radiation Dose Calculator

D = A × (committed effective dose per unit activity)

Health Physics & Dosimetry

CT Scan Radiation Dose Calculator

D = CTDI_vol × (scan length)

Health Physics & Dosimetry

Diagnostic X-Ray Dose Calculator

D = K × (kVp)^2 × (mAs) / (d^2)

Health Physics & Dosimetry

Background Radiation Calculator

D_bg = D_cosmic + D_terrestrial + D_internal + D_medical

Health Physics & Dosimetry

Radiation Risk Calculator

Risk = E × 0.05 (for cancer), or 0.001 (for heritable)

Health Physics & Dosimetry

Equivalent Dose Calculator

H = w_R D

Health Physics & Dosimetry

Collective Dose Calculator

S = Σ E_i N_i

Health Physics & Dosimetry

Effective Dose Calculator

E = Σ w_T H_T

Nuclear Thermodynamics

Reactor Heat Removal Rate

Q_rem = Σ (flow × cp × ΔT) over all channels

Nuclear Thermodynamics

Nuclear Plant Heat Rate

HR = (thermal input) / (net electrical output)

Nuclear Thermodynamics

Rankine Cycle Efficiency

η_R = (h₁ - h₂) / (h₁ - h₄)

Nuclear Thermodynamics

Steam Generator Efficiency

η_SG = (heat transferred) / (heat available from primary)

Nuclear Thermodynamics

Reactor Cooling Rate

Q = ṁ c_p ΔT

Nuclear Thermodynamics

Thermal Efficiency of Nuclear Power Plant

η = 1 - T_c / T_h

Fusion Engineering

Fusion Reactor Efficiency

η_f = (P_electric) / (P_fusion)

Fusion Engineering

Plasma Confinement Time

τ_E = (energy stored) / (energy loss rate)

Fusion Engineering

Fusion Power Density

S = n² ⟨σv⟩ E / 4

Fusion Engineering

Triple Product Formula

n T τ_E

Fusion Engineering

Fusion Gain Factor (Q)

Q = P_fusion / P_input

Fusion Engineering

Plasma Pressure

p = n k T

Fusion Engineering

Plasma Beta

β = p / (B² / (2 μ₀))

Fusion Engineering

Fusion Reaction Energy

E = Δm c²

Fusion Engineering

Lawson Criterion

n τ > 10²⁰ m⁻³ s (for D-T)

Nuclear Fuel Cycle

Nuclear Waste Heat Generation

Q_waste = Σ P_i (1 - e^{-λ_i t})

Nuclear Fuel Cycle

Fuel Inventory Calculator

I = V × ρ × (1 - porosity) × enrichment

Nuclear Fuel Cycle

Nuclear Fuel Efficiency

NFE = (energy produced) / (energy in fuel)

Nuclear Fuel Cycle

Uranium Consumption Rate

UCR = P / (η × F), where P = thermal power, F = energy per gram

Nuclear Fuel Cycle

Fuel Pellet Density

ρ = mass / (π r² h)

Nuclear Fuel Cycle

Spent Fuel Activity

A_sf = Σ A_i e^{-λ_i t}

Nuclear Fuel Cycle

Fuel Cycle Cost Calculator

FCC = (enrichment cost + fabrication cost + reprocessing cost + disposal cost) / (electricity generated)

Nuclear Fuel Cycle

Fuel Residence Time

t_res = (fuel mass) / (burnup rate)

Nuclear Fuel Cycle

Fuel Utilization Factor

U = (burnup × initial HM) / (initial fissile mass)

Nuclear Decay

Radioactive Dating Formula

t = (1/λ) ln(1 + D/P)

Nuclear Decay

Carbon-14 Dating Calculator

t = ln( (N₀/N) ) / λ

Nuclear Decay

Radioisotope Activity Calculator

A = λ N

Nuclear Decay

Mean Lifetime Formula

τ = 1 / λ

Nuclear Decay

Parent-Daughter Decay Calculator

N_daughter = N_parent(0) (1 - e^{-λt})

Nuclear Decay

Number of Half-Lives Calculator

n = t / T_{1/2}

Nuclear Decay

Remaining Activity Calculator

A = A₀ e^{-λt}

Nuclear Decay

Decay Percentage Calculator

Fraction_remaining = e^{-λt}

Nuclear Decay

Decay Constant Calculator

λ = ln(2) / T_{1/2}

Radiation Protection

Radiation Weighting Factor

w_R = 1 for photons, 2 for protons, 5 for neutrons, 20 for alpha

Radiation Protection

ALARA Radiation Calculator

No single formula; principle of optimisation

Radiation Protection

Occupational Radiation Dose Limit

L = min(50 mSv, 20 mSv over 5 years)

Radiation Protection

Annual Radiation Exposure

E_annual = Σ (D_i × t_i)

Radiation Protection

Radiation Safety Distance Calculator

d = sqrt( (A × Γ) / (D_rate_limit) )

Radiation Protection

Radiation Exposure Time Calculator

t = (D_limit - D_bg) / D_rate

Radiation Protection

Radiation Dose Reduction

DR = 1 - (I/I₀)

Radiation Protection

Lead Shield Thickness

x = ln(I₀/I) / μ

Radiation Protection

Neutron Shielding Thickness

φ = φ₀ e^{-Σ x}

Radiation Protection

Gamma Shielding Thickness

I = I₀ e^{-μx}

Reactor Physics

Reactor Core Volume

V = π r² h

Reactor Physics

Resonance Escape Probability

p = exp(-Σ_i N_i I_i / ξ Σ_s)

Reactor Physics

Neutron Absorption Probability

P_a = Σ_a / Σ_t

Reactor Physics

Reactor Efficiency

η = P_out / P_fission

Reactor Physics

Multiplication Factor Calculator

k = (Production) / (Absorption + Leakage)

Reactor Physics

Reactor Shutdown Margin

SDM = ρ_rod - ρ_excess

Reactor Physics

Control Rod Worth

Δρ = ρ_without - ρ_with

Reactor Physics

Excess Reactivity

ρ = (k - 1) / k

Reactor Physics

Neutron Leakage Probability

P_L = 1 - (1 / k)

Reactor Physics

Thermal Neutron Velocity

v = sqrt(2E/m)

Reactor Physics

Reactor Power Density

P/V = Σ_f Φ E_f

Reactor Physics

Delayed Neutron Fraction

β = Σ β_i

Reactor Physics

Prompt Neutron Lifetime

l = l_p / (1 - β)

Reactor Physics

Four Factor Formula

k = η × ε × p × f

General

Nuclear Fuel Enrichment

Enrichment (%) = (Mass of Fissile Isotope / Total Mass of Element) * 100

General

Q-value of a Nuclear Reaction

Q = (Σm_reactants - Σm_products) * c²

General

Coolant Temperature Rise (Reactor Core)

ΔT = P / (mdot * Cp)

General

Nuclear Reactor Thermal Power

P = mdot * Cp * ΔT

Radiation

Secular Equilibrium Activity Ratio

A_daughter = A_parent (at secular equilibrium)

Radiation

Half-Value Layer

HVL = ln(2) / μ

Radiation

Shielding Exponential Attenuation

I = I0 * e^(-μx)

Radiation

Inverse Square Law for Radiation Intensity

I2 = I1 * (d1/d2)²

Radiation

Absorbed Dose Rate

Ḋ = D / t

Radiation

Radiation Dose Equivalent (Sievert)

H = D * WR

Radiation

Curie to Becquerel Conversion

A(Bq) = A(Ci) * 3.7 * 10^10

Radiation

Specific Activity

SA = A / m = (λ * NA) / M

Radiation

Radioactive Activity

A = λ * N

Reactor Physics

Neutron Generation Time

Λ = l / k_eff

Reactor Physics

Conversion Ratio (Breeding)

CR = Fissile Atoms Produced / Fissile Atoms Consumed

Reactor Physics

Neutron Diffusion Length

L = sqrt(D / Σa)

Reactor Physics

Six-Factor Formula (Infinite Multiplication Factor)

k∞ = η * f * p * ε * P_FNL * P_TNL

Reactor Physics

Critical Mass (Bare Sphere, Simplified)

M_c ≈ (4/3)*π*R_c³*ρ

Reactor Physics

Fuel Burnup

BU = Energy Produced / Mass of Fuel

Reactor Physics

Thermal Utilization Factor

f = Σa_fuel / Σa_total

Reactor Physics

Neutron Mean Free Path

λ = 1 / Σ

Reactor Physics

Macroscopic Cross Section

Σ = N * σ

Reactor Physics

Reactivity

ρ = (k_eff - 1) / k_eff

Reactor Physics

Reactor Period

T = τ / (k_eff - 1)

Reactor Physics

Effective Multiplication Factor (k-eff)

k_eff = (neutrons produced) / (neutrons lost, one generation)

Reactor Physics

Neutron Flux

φ = n * v

Radiation

Half-Life Relation

t₁/₂ = ln2 / λ

Radiation

Radioactive Decay Law

N = N₀e^(−λt)

Reactor Physics

Nuclear Binding Energy

E_B = Δmc²

Reactor Physics

Mass-Energy Equivalence

E = mc²