Decay Heat Calculator
Calculate decay heat with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Calculator
Adjust values & calculateDecay Heat vs Time After Shutdown
Formula
Where P(t) is the decay heat at time t after shutdown, P0 is the operating thermal power, t is the time after shutdown in seconds, and ts is the total operating time before shutdown in seconds. The constant 0.066 represents approximately 6.6% of operating power.
Last reviewed: December 2025
Worked Examples
Example 1: PWR Reactor One Day After Shutdown
Example 2: Spent Fuel Pool Heat Load
Background & Theory
The Decay Heat Calculator applies the following established principles and formulas. Physics is the fundamental natural science concerned with matter, energy, and the interactions between them. Classical mechanics, founded on Newton's three laws of motion, provides the framework for analyzing the motion of objects. The first law states that an object remains at rest or in uniform motion unless acted upon by a net external force. The second law quantifies this relationship: F = ma, where force equals mass times acceleration in SI units of newtons (N = kgยทm/sยฒ). The third law establishes that every action produces an equal and opposite reaction. Kinematics describes motion without reference to its causes. The four fundamental equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t: v = u + at, s = ut + ยฝatยฒ, vยฒ = uยฒ + 2as, and s = ยฝ(u + v)t. These assume constant acceleration and are foundational for solving projectile motion, free fall, and linear dynamics problems. Energy conservation underpins much of physics. Kinetic energy is KE = ยฝmvยฒ, where m is mass in kilograms and v is speed in meters per second. Gravitational potential energy is PE = mgh, where g โ 9.81 m/sยฒ near Earth's surface and h is height in meters. The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W = ฮKE. Electricity and circuits rely on Ohm's law: V = IR, where voltage V is in volts, current I in amperes, and resistance R in ohms. Electrical power is P = IV = IยฒR = Vยฒ/R, measured in watts. Wave mechanics connects frequency f, wave speed v, and wavelength ฮป through f = v/ฮป, with frequency in hertz (Hz). Pressure is defined as force per unit area, P = F/A, in pascals (Pa = N/mยฒ). The ideal gas law PV = nRT links pressure, volume, moles n, the gas constant R = 8.314 J/(molยทK), and absolute temperature in kelvin. Gravitational force between two masses follows Newton's law of universal gravitation: F = Gmโmโ/rยฒ, where G = 6.674ร10โปยนยน Nยทmยฒ/kgยฒ is the gravitational constant.
History
The history behind the Decay Heat Calculator traces back through the following developments. The history of physics spans over two millennia, beginning with the natural philosophy of ancient Greece. Aristotle (384โ322 BCE) proposed that all matter consisted of four elements and that objects moved toward their natural place, with heavier objects falling faster than lighter ones. While largely incorrect, his systematic approach to explaining nature dominated Western thought for nearly 2,000 years. The Scientific Revolution overturned Aristotelian physics. Galileo Galilei (1564โ1642) performed groundbreaking experiments on inclined planes and falling bodies, demonstrating that all objects fall with the same acceleration regardless of mass, and established the principle of inertia. His use of mathematics to describe motion was revolutionary. Isaac Newton synthesized these developments in his landmark Principia Mathematica (1687), laying out the three laws of motion and the law of universal gravitation. Newton's framework unified terrestrial and celestial mechanics, explaining planetary orbits with the same equations governing a falling apple. His calculus provided the mathematical language for expressing rates of change. The 19th century brought two major theoretical achievements. James Clerk Maxwell formulated his equations of electromagnetism between 1861 and 1862, unifying electricity, magnetism, and optics, and predicting the existence of electromagnetic waves traveling at the speed of light. Thermodynamics was developed by Carnot, Clausius, and Kelvin, establishing the laws governing heat, work, and entropy. The 20th century produced two revolutions that fundamentally altered the classical picture. Albert Einstein published the special theory of relativity in 1905, showing that space and time are not absolute but relative to the observer, and that mass and energy are equivalent via E = mcยฒ. His general theory of relativity in 1915 reinterpreted gravity as the curvature of spacetime. Simultaneously, quantum mechanics emerged from the work of Planck, Bohr, Heisenberg, and Schrรถdinger, revealing that at atomic scales energy is quantized and particles exhibit wave-particle duality. These developments culminated in the Standard Model of particle physics, which describes all known fundamental particles and three of the four fundamental forces.
Frequently Asked Questions
Formula
P(t)/P0 = 0.066 * [t^(-0.2) - (ts + t)^(-0.2)]
Where P(t) is the decay heat at time t after shutdown, P0 is the operating thermal power, t is the time after shutdown in seconds, and ts is the total operating time before shutdown in seconds. The constant 0.066 represents approximately 6.6% of operating power.
Worked Examples
Example 1: PWR Reactor One Day After Shutdown
Problem: A 3000 MWt pressurized water reactor has operated for 365 days. Calculate the decay heat 1 day (86,400 seconds) after shutdown.
Solution: P0 = 3000 MW = 3e9 W\nts = 365 * 86400 = 31,536,000 s\ntsd = 86,400 s\nP/P0 = 0.066 * (86400^-0.2 - (31536000+86400)^-0.2)\n= 0.066 * (0.09120 - 0.02776)\n= 0.066 * 0.06344 = 0.004187 = 0.419%\nDecay heat = 0.004187 * 3e9 = 12.56 MW
Result: Decay Heat: 12.56 MW (0.419% of operating power) after 1 day
Example 2: Spent Fuel Pool Heat Load
Problem: Calculate decay heat from a 1000 MWt reactor fuel bundle 30 days after shutdown, operated for 3 years (1095 days).
Solution: P0 = 1000 MW = 1e9 W\nts = 1095 * 86400 = 94,608,000 s\ntsd = 30 * 86400 = 2,592,000 s\nP/P0 = 0.066 * (2592000^-0.2 - (94608000+2592000)^-0.2)\n= 0.066 * (0.04542 - 0.02488)\n= 0.066 * 0.02054 = 0.001356 = 0.136%\nDecay heat = 0.001356 * 1e9 = 1.356 MW
Result: Decay Heat: 1.356 MW (0.136% of operating power) after 30 days
Frequently Asked Questions
What is decay heat and why is it important for nuclear reactor safety?
Decay heat is the thermal energy produced by the radioactive decay of fission products and actinides in nuclear fuel after the fission chain reaction has been stopped. Even after a reactor is shut down by inserting control rods and stopping the fission process, the accumulated fission products continue to emit beta particles, gamma rays, and alpha particles as they undergo radioactive decay, generating significant heat. Immediately after shutdown, decay heat is approximately 6 to 7 percent of the full operating power, which for a 3000 MW thermal reactor means about 200 MW of heat must still be removed. This is why nuclear reactors require robust and redundant cooling systems that function even during shutdown conditions.
How does the Way-Wigner formula estimate decay heat after reactor shutdown?
The Way-Wigner formula is an empirical approximation that estimates the fraction of operating power remaining as decay heat at time t after shutdown. The formula is P(t)/P0 equals 0.066 times the quantity t to the negative 0.2 power minus (ts plus t) to the negative 0.2 power, where ts is the operating time before shutdown and t is the time after shutdown. The factor 0.066 represents approximately 6.6 percent of operating power, and the negative 0.2 exponent captures the characteristic slow decay of fission product activity. This formula is remarkably simple yet provides reasonable estimates (within about 20 percent) for shutdown times from about one second to one hundred million seconds after a long period of reactor operation.
How does operating time before shutdown affect the decay heat level?
The operating time before shutdown significantly affects the initial decay heat level and its long-term evolution. Longer operating times allow more fission products to accumulate, including longer-lived isotopes that contribute to decay heat for extended periods. A reactor that has operated for one year produces more decay heat at any given time after shutdown than one operated for just one month, because the long-lived fission products (with half-lives of months to years) have had more time to build up. However, the effect of operating time diminishes for short-lived isotopes, which reach secular equilibrium relatively quickly. For operating times exceeding about two years, the decay heat profile becomes largely independent of operating duration for the first few days after shutdown.
What role did decay heat play in the Fukushima nuclear accident?
The Fukushima Daiichi accident in March 2011 was fundamentally a decay heat removal failure. When the earthquake struck, the three operating reactors automatically shut down their fission chain reactions successfully. However, the subsequent tsunami destroyed the backup diesel generators needed to power the emergency cooling systems. Without active cooling, the decay heat (still producing tens of megawatts of thermal power) caused the water in the reactor vessels to boil away, uncovering the fuel. The fuel temperature then rose to the point where zirconium cladding reacted with steam, producing hydrogen gas and causing explosions. This accident dramatically demonstrated that managing decay heat is equally important as controlling the chain reaction in nuclear safety.
How does decay heat change over time after reactor shutdown?
Decay heat decreases monotonically after shutdown but follows a complex multi-exponential decline rather than a simple single exponential. In the first few seconds, very short-lived fission products (half-lives of fractions of a second) decay rapidly, causing a quick initial drop. Within the first hour, the decay heat drops to about 1.5 percent of operating power. After one day, it is approximately 0.5 percent. After one week, about 0.2 percent. After one month, roughly 0.1 percent. After one year, approximately 0.02 percent. Even years after shutdown, the decay heat from long-lived isotopes like cesium-137 and strontium-90 remains significant and requires continued cooling in spent fuel storage facilities.
What fission products contribute most to decay heat at different times?
The dominant contributors to decay heat change dramatically with time after shutdown. In the first few minutes, short-lived isotopes like neptunium-239 (2.36 days), various iodine isotopes, and noble gas fission products dominate. From hours to days, isotopes such as barium-140, lanthanum-140, iodine-131, and molybdenum-99 are major contributors. From weeks to months, zirconium-95, niobium-95, cerium-144, and ruthenium-103 become important. At one year and beyond, cesium-137 (30 years) and strontium-90 (29 years) dominate, producing roughly equal contributions. For very long-term storage (thousands of years), actinides like plutonium-241 and americium-241 become the primary decay heat sources.
References
Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy