General heat equation in 3D

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.

Conditions:

1. The medium is isotropic.

2. There is no mass transfer or radiation in the system.

Links:

1. Wikipedia — Heat equation.

medium_density

density of the medium.

Symbol:

rho

Latex:

\(\rho\)

Dimension:

mass/volume

medium_specific_isobaric_heat_capacity

mass-specific isobaric heat_capacity of the medium.

Symbol:

c_p

Latex:

\(c_{p}\)

Dimension:

energy/(mass*temperature)

position_vector

Position vector. See distance_to_origin.

Symbol:

r

Latex:

\({\vec r}\)

Dimension:

length

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

temperature

temperature as a function of position_vector and time.

Symbol:

T(r, t)

Latex:

\(T{\left({\vec r},t \right)}\)

Dimension:

temperature

thermal_conductivity

thermal_conductivity as a function of position_vector.

Symbol:

k(r)

Latex:

\(k{\left({\vec r} \right)}\)

Dimension:

power/(length*temperature)

heat_source_density

Volumetric density of heat sources (i.e. power produced per unit volume) as a function of position_vector.

Symbol:

q(r)

Latex:

\(q{\left({\vec r} \right)}\)

Dimension:

power/volume

law

rho * c_p * Derivative(T(r, t), t) = div(k(r) * grad(T(r, t))) + q(r)

Latex:

\[\rho c_{p} \frac{\partial}{\partial t} T{\left({\vec r},t \right)} = \text{div} \, k{\left({\vec r} \right)} \text{grad} \, T{\left({\vec r},t \right)} + q{\left({\vec r} \right)}\]