Question

The conduction equation boundary condition for an adiabatic surface with direction \(n\) being normal to the surface is (a) \(T=0\) (b) \(d T / d n=0\) (c) \(d^{2} T / d n^{2}=0\) (d) \(d^{3} T / d n^{3}=0\) (e) \(-k d T / d n=1\)

Short answer

Answer: The correct boundary condition for an adiabatic surface is \(d T / d n=0\).

Step-by-step solution

Definition of an adiabatic surface

An adiabatic surface is a surface where there is no heat transfer across it. This means that the rate of heat transfer through the surface is zero.

Conduction Equation

The rate of heat transfer by conduction is given by Fourier's law, which states that: \[q = -k \frac{dT}{dn}\] where \(q\) is the rate of heat transfer per unit area, \(k\) is the thermal conductivity of the material, and \(\frac{dT}{dn}\) is the temperature gradient in the normal direction to the surface.

Adiabatic Boundary Condition

The boundary condition for an adiabatic surface is that the rate of heat transfer (\(q\)) through the surface must be zero. Therefore, from Fourier's law, we get: \[0 = -k \frac{dT}{dn}\]

Simplify the Boundary Condition

Since \(k\) is a constant, it can be derived that the temperature gradient in the normal direction to the surface must be zero. Thus, the adiabatic boundary condition for the conduction equation is: \[\frac{dT}{dn} = 0\]

Final Answer

The correct boundary condition for an adiabatic surface is option (b): \[d T / d n=0\]