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Differential Equations Solved in the Thermal Analysis

Cv: Volumetric heat capacity (the product of specific heat and density)

θ: Temperature

q: Heat flux

Q: Heat density

λ: Thermal conductivity

 

 

Femtet Solver "Watt" solves Equations (1) and (2) to obtain the distributions of θ and q.

 

Expressions (1) and (2) express the following.

 

Equation (1): law of conservation of energy that holds in the solid with volumetric specific heat Cv

Equation (2): Fourier's law that holds in the solid with thermal conductivity λ

 

 

Governing Equation

 

Solver "Watt" solves Equation (3), which is derived from Equations (1) and (2).

 

 

Cv: Volumetric heat capacity (the product of specific heat and density)

θ: Temperature

λ: Thermal conductivity

 

In steady state, the left side of Equation (3) is equal to zero.

 

For transient analysis, fully-implicit method is used.

 

Once θ is obtained, q can be obtained from Equations (2).

 

 

Some boundary conditions must be given to solve Equation (3).

The typical boundary conditions:

 

Temperature: The temperature θ of the boundary is fixed at the entered value.

Adiabatic: The normal component of heat flux q (－λ∇θ) on the boundary is equal to zero.

Heat flux: The normal component of heat flux q (－λ∇θ) on the boundary is set at the entered value.

 

 

By applying the heat transfer: convection or radiation boundary condition, the relationship of ⊿θ and the amount of heat radiation can be set where ⊿θ is the difference between surface temperature θ and ambient temperature θroom, and amount of heat radiation is in reversed normal direction to the heat flux boundary.