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A heat generation rate specifies the power deposited into microscopic thermal degrees of freedom, usually through conversion of electrical, chemical or mechanical energy. Its unit is the watt. The term is used as a source in a thermal energy balance; it does not imply creation of total energy.

A resistor supplied with current $I$ has electrical power $I^{2}R$ converted by Joule heating. If that power is deposited approximately uniformly in a body of volume $V$, the volumetric heat generation rate is $q^{\prime \prime \prime }=I^{2}R/V$. A steady body can export the same total power through heat transfer and retain constant temperature. If it is approximately isolated for a short interval, the deposited energy instead raises its internal energy, giving $C\dot{T}=P$ when its heat capacity is approximately constant.

Mixed oxide fuel begins as a specified mixture of uranium and plutonium oxides. A fuel pin develops strong radial temperature gradients during operation. The components and oxygen need not remain spatially uniform: their chemical potentials depend on temperature, local composition and oxygen-to-metal ratio.

Actinide redistribution in oxide fuel can involve thermally driven solid-state diffusion and evaporation–condensation transport across pores or a central cavity. Oxygen redistributes as well, changing local oxidation states and the relative thermodynamic tendencies of uranium- and plutonium-bearing species. Pore migration, grain growth and central-void formation couple compositional redistribution to fuel restructuring. Differential local burnup also changes composition over time.

The direction and magnitude of plutonium migration cannot be inferred from a temperature gradient alone. In particular irradiated fast-reactor pins, enrichment around a central cavity has been observed, but the result depends on stoichiometry and operating history. M. Bober and colleagues investigated thermal-diffusion and evaporation–condensation effects using autoradiographs of sections of fuel irradiated in RAPSODIE.^[ref]

Redistribution changes the radial heat generation rate and material properties. Concentrating more fissile material near an already hot region can raise local heat generation; changes in thermal conductivity and local melting behavior modify the temperature response. Gas release, pore formation and fuel–cladding contact can also change thermal resistance and mechanical loading. A spatially averaged composition can therefore underestimate a local temperature or chemical interaction.

Segregated fission products add another concern: mobile species can accumulate at colder interfaces or react with the cladding, while insoluble products form separate phases. Fuel performance must therefore track composition, temperature and microstructure together, rather than treating segregation as merely a cosmetic change in an otherwise fixed fuel.

Assume a stationary temperature profile, spatially uniform mass density and radioactive heat generation rate, constant thermal conductivity, isotropic solid thermal conduction, and blackbody emission into effectively cold surroundings. The volumetric source isTotal radiogenic power equals the surface blackbody radiation loss. The Stefan–Boltzmann law givesFor steady radiogenic heating of a conducting sphere, the interior heat equation isRegularity at the centre removes the $1/r^2$ integration term and yields $dT/dr=-q_{v}r/(3\kappa )$. Matching $T(R)=T_S$ givesThe high central temperature is compatible with a cold surface because finite thermal conductivity requires a large temperature gradient to carry the accumulated interior heat. A nonzero background radiation temperature $T_{\mathrm{bg}}$ replaces $T_S^4$ by $T_S^4-T_{\mathrm{bg}}^4$ in the surface balance. The calculation neglects convection, stored transient heat, changing radioactive abundances, phase changes and any substantial external irradiation.

For constant volumetric heat generation rate $q_v$ and thermal conductivity $\kappa$, spherical steady thermal conduction obeys $\kappa r^{-2}(r^2{T}^{\prime }{)}^{\prime }+q_v=0$. Regularity at the centre and a specified surface temperature give the displayed parabolic profile. For a black surface radiating into negligible background radiation, global energy conservation gives ${\sigma }_{\mathrm{SB}}{T}_S^4=q_{v}R/3$. The central excess scales as $R^2$, while the surface temperature scales only as $R^{1/4}$.

Thermal power generated per unit volume. Its integral over a region is the region's total heat generation rate.