**Numerical Solutions Applied To Heat
Transfer And Fluid Mechanics Problems**

**Steps in the finite difference solution
of a problem**

*Determine the appropriate mathematical
model for the problem.*

- Conservation equations
- Boundary conditions
- Initial conditions

Example: Transient Heat Conduction

Energy conservation equation

Boundary Conditions

At a surface of a conducting structure either the temperature or the heat flux is specified. A fixed temperature does not represent of any real physical configuration, but can be an appropriate approximation for a structure adjacent to a region with very high conductivity and total heat capacity. Surface heat flux within the structure is expressed in terms of the normal derivative of the temperature. Useful flux boundary conditions include:

Constant surface heat flux,

Adiabatic boundary condition,

Convection,

Convection and radiation

For problems with conduction through two regions with two different materials, fluxes must match at the boundary between the regions,

Initial Conditions

T(x, y, z, t=0) = f(x,y,z)

Example: Fluid Flow with Heat Transfer (2D - Steady State)

Conservation equations:Continuity

Momentum Equations (u is "x" component, v is "y" component)Energy Equation

Boundary Conditions

Inlet | x = 0, | u = U_{in}, |
v = 0, | T = T_{in} |

Walls | y = b, | u = 0, | v = 0, | T = T_{w} |

Symmetry | y = 0, | v = 0, | ||

Outlet | ? |