CFD Computational Fluid Dynamics with heat and mass transfer

1
CFDComputational Fluid Dynamics (with heat and
mass transfer)

• MECH3950

2
Methods of prediction

• Experimental investigation
• full scale
• expensive and often impossible
• measurement errors
• on a small scale model
• simplified
• difficult to extrapolate results
• measurement errors
• Theoretical calculation
• analytical solutions
• exist only for a few cases
• sometimes complex
• numerical solutions
• for almost any problem

3
Modeling vs. experimentation

• Advantages of modeling
• cheaper
• more complete information
• can handle any degree of complexity as long as
• Disadvantages of modeling
• deals with a mathematical description not with
  reality
• mathematical description can be inadequate
• multiple solutions can exist
• and the winner is

4
Mathematical description of transfer processes

• Conservation principle leads to a general
  differential equation

5
Nature of coordinates

• with respect to xj
• 1D, 2D, 3D
• with respect to t
• steady
• unsteady(transient, time dependent)
• one-way coordinates gt parabolic problem
• two-way coordinates gt elliptic problem

6
Discretization methods

• analytical solution gives a function (continuous)
• numerical solution gives a set of numbers
  (discrete)

7
Example steady 1D heat conduction
8
Integration
9
Discretization
10
Discretization equation
conductances
heat generation
11
Boundary conditions

• 1. Given boundary temperature
• simple
• 2. Given boundary heat flux
• a) as a constant
• b) specified via a heat transfer coefficient
  and the temperature of the surrounding fluid

12
Boundary conditions
13
Boundary conditions
14
Boundary conditions

• if qB specified as a constant then

15
Boundary conditions

• if qB specified via a heat transfer h
  coefficient and the temperature of the
  surrounding fluid Tf

16
Solution of the linear algebraic equations

• Possibilities
• Iteration
• Gaussian elimination
• Tri-Diagonal-Matrix-Algorithm TDMA

change of notation
17
TDMA
what happens when a boundary temperature is
given ?
18
TDMA
19
TDMA
20
TDMA algorithm

• CalculateP1b1/a1 and Q1d1/a1
• Use recurrence relations to get Pi and Qi for
  i2,3N.
• Set TN QN
• Use recurrence relations to getTi Pi Ti1 Qi
  for iN-1, N-23, 2, 1
• to obtain
• TN-1, TN-2, T3, T2, T1.