Fourier's Law for Insulation Formula

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Fourier's Law for Insulation Formula

HVAC Design and Engineering
Heat Transfer

Fourier's Law for Insulation Formula

Alternative resource for calculating heat loss or gain: Heat Loss from Ducts Equations and Calculator

Steady state, one-dimensional heat flow through insulation systems is governed by Fourier’s law:

Fourier's Law Equation

Q = –kA dT/dx

Where:

Q = rate of heat flow, Btu/h
A = cross-sectional area normal to heat flow, ft²
k = thermal conductivity of insulation material, Btu/h·ft·°F
dT/dx = temperature gradient, °F/ft

For flat geometry of finite thickness, the equation reduces to:

Q = kA(T₁ – T₂)/L

where L is insulation thickness, in ft.

For radial geometry, the equation becomes

Q = kA2(T₁– T₂)/[r₂ ln(r₂/r₁)]

Where
r₂ = outer radius, ft
r₁ = inner radius, ft
A₂ = area of outer surface, ft²

The term r₂ ln( r₂/r₁) is sometimes called the equivalent thickness of the insulation layer. Equivalent thickness is the thickness of insulation that, if installed on a flat surface, would equal the heat flux at the outer surface of the cylindrical geometry.

Heat transfer from surfaces is a combination of convection and radiation. Usually, these modes are assumed to be additive, and therefore a combined surface coefficient can be used to estimate the heat flow to and from a surface:

h_s = h_c + h_r

where

h_s = combined surface coefficient, Btu/h·ft²·°F
h_c = convection coefficient, Btu/h·ft²·°F
h_r = radiation coefficient, Btu/h·ft²·°F

Assuming the radiant environment is equal to the ambient air temperature, the heat loss/gain at a surface can be calculated as

Q = h_sA(T_surf – T_amb)

The radiation coefficient is usually estimated as:

h_r = ε σ ( T⁴_surp - T⁴_amb) / ( T_surp - T_amb)

Where:

ε = surface emittance
σ = Stephen-Boltzmann constant, 0.1712 × 10^-8 Btu/h·ft²·°R⁴

Reference:

ASHRAE Fundamental Handbook, 2019