Heat loss from a pipe

The total heat loss from a pipe is given by:

where , giving:

This assumes steady-state, one-dimensional radial heat transfer with resistances acting in series. The model covers three physical mechanisms: conduction through the pipe wall and insulation, convection at the inner and outer surfaces, and radiation from the outer insulation surface to the surroundings.

Iterative solution required

The radiative coefficient depends on the outer insulation surface temperature , which is itself an output of the calculation. The resistance network must therefore be solved iteratively: assume , compute , solve for , back-calculate from the outer resistance, and repeat until convergence.

Nomenclature

Symbol	Description	Units
	Total heat loss rate	W
	Bulk fluid temperature (inside pipe)	K
	Ambient temperature (surroundings)	K
	Inner pipe wall temperature	K
	Outer pipe wall temperature (pipe–insulation interface)	K
	Outer insulation surface temperature	K
	Inner convective heat transfer coefficient	W/m²·K
	Outer convective heat transfer coefficient	W/m²·K
	Radiative heat transfer coefficient	W/m²·K
	Inner pipe surface area,	m²
	Outer insulation surface area,	m²
	Inner pipe radius	m
	Outer pipe radius (pipe–insulation interface)	m
	Outer insulation radius	m
	Thermal conductivity of pipe wall	W/m·K
	Thermal conductivity of insulation	W/m·K
	Pipe length	m
	Emissivity of insulation outer surface	—
	Stefan–Boltzmann constant,	W/m²·K⁴
	Inner convective resistance	K/W
	Pipe wall conduction resistance	K/W
	Insulation conduction resistance	K/W
	Outer combined (convection + radiation) resistance	K/W

Mechanisms

Conduction

In a solid, the flow of heat by conduction is the result of the transfer of vibrational energy from one molecule to another, and in fluids it occurs in addition as a result of the transfer of kinetic energy. Heat transfer by conduction may also arise from the movement of free electrons, a process which is particularly important with metals and accounts for their high thermal conductivities.

— Coulson and Richardson, Vol. 1

Conduction through plane walls

The rate of heat flow across area over an infinitesimal distance in a material of thermal conductivity is given by Fourier’s law:

INFO

The negative sign indicates that heat flows in the direction of decreasing temperature. Integrating over a finite thickness , where varies linearly with temperature:

where is the arithmetic mean of the thermal conductivities at and .

The ratio represents the thermal resistance per unit area. For multiple layers in series:

Conduction through a tube wall

For a cylindrical geometry, the heat flux is proportional to the local surface area, which increases with radius. The temperature gradient is therefore inversely proportional to radius. At any radius in a tube of length and thermal conductivity :

Separating variables and integrating between and :

For multiple layers (e.g. over a pipe wall and insulation):

Convection

Heat transfer by convection arises from the mixing of elements of fluid. If this mixing occurs as a result of density differences — as, for example, when a pool of liquid is heated from below — the process is known as natural convection. If the mixing results from eddy movement in the fluid, for example when a fluid flows through a pipe heated on the outside, it is called forced convection. It is important to note that convection requires mixing of fluid elements, and is not governed by temperature difference alone as is the case in conduction and radiation.

— Coulson and Richardson, Vol. 1

When a fluid is in contact with a surface at a different temperature, heat is transferred by convection. The rate of heat transfer is described by Newton’s law of cooling:

where is the convective heat transfer coefficient, is the surface area, is the surface temperature, and is the bulk fluid temperature.

Inner convection — fluid to pipe inner wall

where is the inner convective heat transfer coefficient, , is the bulk fluid temperature, and is the pipe inner wall temperature. For turbulent internal flow, is typically estimated using the Dittus–Boelter equation or a suitable Nusselt number correlation.

Outer convection — insulation surface to ambient air

where and is the ambient air temperature.

The outer convective coefficient is not a fixed property — it depends on the pipe geometry, the surface-to-air temperature difference, and the physical properties of the surrounding fluid. For a horizontal pipe losing heat to quiescent air by natural convection, the calculation chain proceeds as follows.

The Grashof number quantifies the ratio of buoyancy to viscous forces:

where is gravitational acceleration, is the thermal expansion coefficient of air, is the pipe outer diameter (the characteristic length), and is the kinematic viscosity of air. Multiplying by the Prandtl number gives the Rayleigh number:

An empirical correlation maps to the Nusselt number , with the form depending on whether the boundary layer is laminar or turbulent. The coefficient is then recovered from:

where is the thermal conductivity of air evaluated at an appropriate film temperature.

Radiation

All materials radiate thermal energy in the form of electromagnetic waves. When this radiation falls on a second body it may be partially reflected, transmitted, or absorbed. It is only the fraction that is absorbed that appears as heat in the body.

— Coulson and Richardson, Vol. 1

Radiation from the outer insulation surface to the surroundings is given by the Stefan–Boltzmann law:

where is the emissivity of the outer insulation surface and W/m²·K⁴.

This expression is appropriate for a pipe radiating to an open environment but should be reconsidered if the pipe is in a confined or high-temperature enclosure.

Linearised radiative coefficient

To incorporate radiation into the resistance network, equation (12) is linearised so the radiative heat flux can be written in the form :

Reference

Ref equation (5-12) in Perry’s Chemical Engineers Handbook

The total outer heat transfer (convection and radiation in parallel, sharing the same surface area and the same driving temperature ) is then:

which yields the outer thermal resistance:

The parallel addition of and is valid provided the convective reference temperature equals the radiative sink temperature, i.e. . This holds in most outdoor and unconfined industrial settings but may not hold near furnaces etc.

Since depends on , which is itself unknown, the solution is implicit and requires iteration (see note at the start of this article).