Heat equation

From Chemepedia

The heat equation describes the temperature field in a flowing continuum. Heat is energy in transit and hence is not a system property. The associated partial differential equation is based on the flow of thermal or internal energy through a system.

General development

We start with the general formulation of the microscopic balance using the Eulerian specification for the transport quantity of internal energy, or

            (1)

where

  • is the mass density
  • is the the transport quantity on a unit mass basis, which in this case is
  • is the convective heat flux due to both bulk flow and molecular transfer mechanisms.
  • , which are is reversible generation/consumption of internal energy due to compression/expansion plus the irreversible generation of internal energy due to viscous dissipation.

Eq. (1) therefore is expressed as

Compact form

The heat equation in the material derivative form can be written compactly as

            (2)

where

  • represents the material derivative for the internal energy
  • is the rate of internal energy addition by heat conduction per unit volume.
  • is the reversible rate of internal energy increase or decrease per unit volume by compression or expansion.
  • is the irreversible rate of internal energy increase per unit volume due to viscous dissipation

Note that Eq. (1) has no internal generation terms due to a chemical reaction or other heat source. Such terms would have to be added to the heat equation if present.

Simplified heat equation in terms of temperature and constant thermal conductivity

Assuming negligible viscous dissipation and a constant thermal conductivity, Eq. (1) can be written in terms of either or . If written in terms of the isochoric heat capacity, the heat equation is

            (3)

where

  • is determined from an equation of state. For an ideal gas,

The heat equation written in terms of the isobaric heat capacity is

            (4)

where

  • is determined from an equation of state. For an ideal gas,

Restricted forms of the heat equation

Incompressible flow

In the incompressible flow approximation, and both Eqs. (2) and (3) reduce to

            (5)

Isobaric flow

In isobaric flow and Eq. (3) reduces to

            (6)

Note that in isobaric flow, the isobaric and isochoric heat capacities are not necessarily the same. The isobaric heat capacity must be used in the above expression.

Conduction in solid, incompressible materials

In the absence of flow, the heat equation is expressed as

            (7)

Common boundary conditions

  • Temperature specified at a boundary,
  • Insulated boundary, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot \mathbf{q}\cdot\hat\mathbf{n} = 0 }
  • Convective heat transfer, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot \mathbf{q}\cdot\hat\mathbf{n}=h\Delta T }