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:

1. There is no straghtforward solution to this equation, and it depends on initial conditions as well.

2. To get a similar equation for the 3-dimensional case, replace the spatial derivative with gradient \(\nabla\).

3. 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:

1. Wikipedia.

position

position, or spatial variable.

Symbol:

x

Latex:

\(x\)

Dimension:

length

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

temperature

temperature as a function of position and time.

Symbol:

T(x, t)

Latex:

\(T{\left(x,t \right)}\)

Dimension:

temperature

medium_density

density of the medium.

Symbol:

rho

Latex:

\(\rho\)

Dimension:

mass/volume

medium_specific_isobaric_heat_capacity

heat_capacity of the medium at constant pressure per unit mass.

Symbol:

c_p

Latex:

\(c_{p}\)

Dimension:

energy/(mass*temperature)

thermal_conductivity

thermal_conductivity of the medium as a function of position.

Symbol:

k(x)

Latex:

\(k{\left(x \right)}\)

Dimension:

power/(length*temperature)

heat_source_density

Density of the rate of heat production by external sources as a function of position and time. See energy_density.

Symbol:

q(x, t)

Latex:

\(q{\left(x,t \right)}\)

Dimension:

energy/volume

law

rho * c_p * Derivative(T(x, t), t) = Derivative(k(x) * Derivative(T(x, t), x), x) + q(x, t)

Latex:

\[\rho c_{p} \frac{\partial}{\partial t} T{\left(x,t \right)} = \frac{\partial}{\partial x} k{\left(x \right)} \frac{\partial}{\partial x} T{\left(x,t \right)} + q{\left(x,t \right)}\]