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

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

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

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

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,

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

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

Problem 2.12

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

Problem 2.23