Temperature Model 1d Instructions

From GeoMod

Temperature diffusion through a linear rod. 1d temperature diffusion equation.

Default setup:

• Boundary conditions:
  1. 30 degrees Celcius on left boundary when t > 0
  2. 5 degrees Celcius in right boundary when t > 0

• Initial Conditions:
  • 5 degrees Celcius for entire rod.

Software

Excel model file: Media:Temp-model-1d.xls

Equations

Governing equation for transient temperature diffusion

Descretized equations:

• Implicit equation

  <math> T^{t+1}_{i} = \frac{T^{t}_{i} + S \cdot T^{t+1}_{i+1} + S \cdot T^{t+1}_{i-1}}{1+2 \cdot S} </math>

  

• Explicit Equation

  <math> K \nabla^2 T = \frac{\partial T}{\partial t} </math>

  

• Where

  <math> S = \frac{K \Delta t}{\Delta x^2} </math>

  

  K = thermal conductivity (W/mK) [1]

  T = temperature (deg. Celcius or Kelvin)

  t = time (s)

  x = distance (m)