General equation in one dimension¶
Heat equation governs heat diffusion, as well as other diffusive processes. It describes the evolution of heat transferred from hotter to colder environments in time and space.
Notes:
There is no straghtforward solution to this equation, and it depends on initial conditions as well.
To get a similar equation for the 3-dimensional case, replace the spatial derivative with gradient \(\nabla\).
Thermal conductivity \(k\) can depend not only on position, but also on local temperature, but this is out of the scope of this law.
Links:
- temperature¶
Temperature as a function of position and time.
- Symbol:
T(x, t)
- medium_density¶
Density of the medium.
- Symbol:
rho- Latex:
\(\rho\)
- medium_specific_isobaric_heat_capacity¶
Heat capacity of the medium at constant pressure per unit mass.
- Symbol:
c_p- Latex:
\(c_p\)
- thermal_conductivity¶
Thermal conductivity of the medium as a function of position.
- Symbol:
k(x)
- heat_source_density¶
Density of the rate of heat production by external sources as a function of position and time.
- Symbol:
q(x, t)
- position¶
Position, or spatial variable.
- Symbol:
x
- time¶
Time.
- Symbol:
t
- law¶
rho * c_p * Derivative(T(x, t), t) = Derivative(k(x) * Derivative(T(x, t), x), x) + q(x, t)- Latex:
- \[\rho c_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k(x) \frac{\partial T}{\partial x} \right) + q(x, t)\]