Formulations in Heat transfer
With QuickField you can analyze linear and nonlinear temperature fields.
Heat-transfer equation for linear problems is:
Planar case:
\[ \frac{\partial}{\partial x} \left( λ_x\frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y}\left( λ_y\frac{\partial T}{\partial y} \right) = -q - c·ρ \frac{\partial T}{\partial t}, \]
Axisymmetric case: \[ \frac{1}{r} \frac{\partial}{\partial r} \left( λ_r r\frac{\partial T}{\partial r} \right) + \frac{\partial}{\partial z}\left( λ_z\frac{\partial T}{\partial z} \right) = -q - c·ρ \frac{\partial T}{\partial t}, \]
Heat-transfer equation for nonlinear problems is:
Planar case:
\[ \frac{\partial}{\partial x} \left( λ(T)\frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y}\left( λ(T)\frac{\partial T}{\partial y} \right) = -q(T) - c(T)·ρ \frac{\partial T}{\partial t}, \]
Axisymmetric case: \[ \frac{1}{r} \frac{\partial}{\partial r} \left( λ(T) r \frac{\partial T}{\partial r} \right) + \frac{\partial}{\partial z}\left( λ(T)\frac{\partial T}{\partial z} \right) = -q(T) - c(T)·ρ \frac{\partial T}{\partial t}, \]
where:
T - temperature;
t - time;
λx(y,r,z) - components of heat conductivity tensor;
λ(T) - heat conductivity as a function of temperature approximated by cubic spline (anisotropy is not supported in nonlinear case);
q(T) - volume power of heat sources, in linear case - constant, in nonlinear case - function of temperature approximated by cubic spline;
c(T) - specific heat, in nonlinear case - function of temperature approximated by cubic spline;
ρ - density of the substance.
In linear case all parameters are constants within each block of the model.