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Fourier's Law, The Conduction Rate Equation

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Fourier's law is the cornerstone of conduction heat transfer, and its key features are summarized as follows:
•   It is not an expression that may be derived from first principles; it is instead a generalization based on experimental evidence.
•   It is an expression that defines an important material property, the thermal conductivity.
•   Fourier's law is a vector expression indicating that the heat flux is normal to an isotherm and in the direction of decreasing temperature.
•   Fourier's law applies for all matter, regardless of its state (solid, liquid, or gas).

The conduction rate equation is introduced in the conduction lesson here, but will be explored in more detail in this lesson. Fourier's law is phenomenological. For example, consider the steady-state conduction experiment: A cylindrical rod of known material is insulated on its lateral surface, while its end faces are maintained at different temperatures, with T₁ > T₂. The temperature difference causes conduction heat transfer in the positive x direction. The heat transfer rate q_x can be measured. It is of interest to determine how q_x depends on the following variables: ΔT, the temperature difference; Δx, the rod length; and A, the cross-sectional area.

Imagine first holding ΔT and Δx constant and varying A. If this is done, it is found that q_x is directly proportional to A. Similarly, holding ΔT and A constant, it may be observed that q_x varies inversely with Δx. Finally, holding A and Δx constant, it is found that q_x is directly proportional to ΔT. The collective effect is then:

qx  ∝  A

ΔT Δx

Regardless of changing the material (e.g., from a metal to a plastic), the above proportionality remains valid. However, for equal values of A, Δx, and ΔT, the value of q_x would be smaller for the plastic than for the metal. This suggests that the proportionality may be converted to an equality by introducing a coefficient that is a measure of the material behavior. Hence, it is written:

qx  =  kA

ΔT Δx

where k is the thermal conductivity [W/(mK)]. Evaluating the above expression in the limit as Δx → 0, the heat rate becomes:

(Eq1)

qx  =  −kA dT dx

In thermodynamics q_x may be expressed as a capital Q with a dot over it. For the heat flux:

(Eq2)

qx"  =   qx A   =  −k dT dx

The minus sign is necessary because heat is always transferred in the direction of decreasing temperature. The minus sign gives a direction of the heat transfer from a higher temperature to a lower temperature region. Often the gradient is evaluated as a temperature difference divided by a distance when an estimate has to be done if a mathematical or numerical solution is not available.

Fourier's law, as written in the above equation, implies that the heat flux is a directional quantity. In particular, the direction of q_x" is normal to the cross-sectional area A. Or, more generally, the direction of heat flow will always be normal to an isothermal surface. The following figure illustrates the direction of heat flow q_x" in a plane wall for which the temperature gradient dT/dx is negative. From the above equation, it follows that q_x" is positive. Note that the isothermal surfaces are planes normal to the x direction.

Recognizing that the heat flux is a vector quantity, a more general statement of Fourier's law can be written:

(Eq3)

qx"  =  −k∇T  =  −k (i ∂T ∂x   +  j ∂T ∂y   +  k ∂T ∂z )

where T(x, y, z) is the scalar temperature field. It is implicit in the above equation that the heat flux vector is in a direction perpendicular to the isothermal surfaces. An alternative form of Fourier's law is therefore:

(Eq4)

qn"  =  −k ∂T ∂n

where q_n" is the heat flux in a direction n, which is normal to an isotherm, as shown in the following figure. The heat transfer is sustained by a temperature gradient along n. Note also that the heat flux vector can be resolved into components such that, in Cartesian coordinates, the general expression for q" is:

(Eq5)

q" = iqx" + jqy" + kqz"

where, from Eq1 it follows that:

(Eq6a)

qx"  =  −k ∂T ∂x

(Eq6b)

qy"  =  −k ∂T ∂y

(Eq6c)

qz"  =  −k ∂T ∂z

Each of these expressions relates the heat flux across a surface to the temperature gradient in a direction perpendicular to the surface. It is also implicit in Eq1 that the medium in which the condition occurs is isotropic. For such a medium the value of the thermal conductivity is independent of the coordinate direction.