In a one-dimensional heat conduction problem, the boundary condition δT/δx = 0 corresponds to what type of surface?

In a one-dimensional heat conduction problem, the boundary condition δT/δx = 0 corresponds to what type of surface?

• A. A convective boundary with ambient fluid.
• B. A surface held at a fixed temperature.
• C. A radiative boundary.
• D. A perfectly insulated surface.

Answer: (D)

A zero temperature gradient at a boundary means no heat can cross that boundary. In one-dimensional conduction, Fourier’s law says the heat flux q = -k dT/dx. If dT/dx is zero at the surface, then q is zero, so there is no conduction through the surface. That is the hallmark of a perfectly insulated (adiabatic) surface. The other scenarios involve nonzero heat transfer: a convective boundary transfers heat with the ambient fluid and typically has a finite gradient; a fixed-temperature surface sets the temperature at the boundary but does not require the gradient to vanish; a radiative boundary involves heat exchange according to radiation laws and depends on temperature to the fourth power.

A zero temperature gradient at a boundary means no heat can cross that boundary. In one-dimensional conduction, Fourier’s law says the heat flux q = -k dT/dx. If dT/dx is zero at the surface, then q is zero, so there is no conduction through the surface. That is the hallmark of a perfectly insulated (adiabatic) surface.

The other scenarios involve nonzero heat transfer: a convective boundary transfers heat with the ambient fluid and typically has a finite gradient; a fixed-temperature surface sets the temperature at the boundary but does not require the gradient to vanish; a radiative boundary involves heat exchange according to radiation laws and depends on temperature to the fourth power.