What is Heat Transfer?

Fourier SeriesConsider a set of eigenfunctions n that are orthogonal, where orthogonality is defined as

for m nAn arbitrary function f(x) can be expanded as series of these orthogonal eigenfunctions

or

Due to orthogonality, we thus know

all other nAmm integrate to zero because m nThus, the constants in the Fourier series are

AME 60634 Int. Heat Trans.D. B. Go # Cartesian Sturm-LiouvilleCharacteristic Value Problemp(x) = 1; q(x) = 0; w(x) = 1

homogeneous B.C.

After Applying Final B.C.Typical B.C.

DirichletNeumannRobinAME 60634 Int. Heat Trans.D. B. Go # Cartesian Sturm-Liouville

Kakac & YennerHeat Conduction, 3rd Ed.AME 60634 Int. Heat Trans.D. B. Go # Cylindrical Sturm-LiouvilleCharacteristic Value Problemp(r) =r; q(r) = 2/r; w(r) = r

homogeneous B.C.

After Applying Final B.C.Typical B.C.

DirichletNeumannRobin

AME 60634 Int. Heat Trans.D. B. Go # Cylindrical Sturm-Liouville

homogeneous B.C.

Typical B.C.

DirichletNeumannRobinSpecial B.C. case: a = 0, b = r0After Applying Final B.C.

AME 60634 Int. Heat Trans.D. B. Go # Cylindrical Sturm-LiouvilleKakac & YennerHeat Conduction, 3rd Ed.

AME 60634 Int. Heat Trans.D. B. Go # Inhomogeneous BC to Homogeneous BC

+=

+AME 60634 Int. Heat Trans.D. B. Go #