1. Heat Diffusion Equation, Boundary Conditions and Initial Conditions
Heat Transfer
Heat Diffusion Equation, Boundary Conditions and Initial Conditions

2. The Heat Diffusion Equation
Objective: To determine T = f (x, y, z, t) for given b.c.’s and i.c.
Homogeneous medium in 3-D Cartesian coordinates
Differential control volume, dx, dy, dz for conduction analysis in 3-D Cartesian coordinates
qz+dz qz
qy+dy qy dz
qx+dx qx z y x dy
same for y and z directions dx

3. Conservation of Energy:
Heat diffusion equation in 3-D Cartesian coordinates, for constant k
For 1-D, steady state (T/t = 0) heat conduction with no energy generation :
Which means that , or qx’’ = Constant
where

4. Heat equation in cylindrical coordinates (r, Φ, z)
Heat equation in spherical coordinates (r, Φ, θ) x y z r Φ
T (r, Φ, z)
See Figure 2.9, p.64 for differential control volume x y z r Φ
T (r, Φ, z) θ
See Figure 2.10, p.65 for differential control volume

5. For a cylinder, the heat components in the radial, circumferential, and axial directions are respectively,
For a sphere, the heat flux components in the radial, polar, and azimuth directions are respectively,

6. Boundary and Initial Conditions for Heat Diffusion
Boundary conditions are the physical conditions at the boundaries of a medium or system, which can be expressed in mathematical forms. Examples are:
Constant surface temperature (Ts) - Dirichlet
@ x = 0 , T = Ts
T (0, t) = Ts
Constant surface heat flux (qs”) - Neumann
a) Finite heat flux
@ x = 0
b) Adiabatic or Insulated Surface x Ts
T (x, t)
qs” x T x

7. Convection surface condition - Cauchy
@ x = 0
Since the heat diffusion equation is 2nd order in spatial coordinates, two boundary conditions must be specified for each coordinate needed to describe a system.
The heat equation in also 1st order in time, and one initial condition (usually at t = 0) must be specified in order to solve the diffusion equation.
T∞,h
T (x, t) x

8. Problem 2.12

9. Problem 2.14
A solid cylinder of length 0.1 m and diameter 25 mm is well insulated on its side, while its end faces are maintained at temperatures of 100°C and 0°C. What is the rate of heat transfer through the rod if it is made of:
(a) pure copper
(b) aluminum alloy 2024-T6
(c) AISI 302 stainless steel
(d) silicon nitride
(e) wood (oak)
(f) magnesia, 85%
(g) pyrex

10. Problem 2.23