Heat Transfer|L2_Introduction on Heat Conduction

Partial different equation of heat conduction

∂∂x(λ∂T∂x)+∂∂y(λ∂T∂y)+∂∂z(λ∂T∂z)+Φ˙=ρCp∂T∂t\frac{\partial}{\partial x}\left(\lambda \frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left(\lambda \frac{\partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} ∂x∂​(λ∂x∂T​)+∂y∂​(λ∂y∂T​)+∂z∂​(λ∂z∂T​)+Φ˙=ρCp​∂t∂T​

1r∂∂r(λr∂T∂r)+1r2∂∂θ(λ∂T∂θ)+∂∂z(λ∂T∂z)+Φ˙=ρCp∂T∂t\frac{1}{r} \frac{\partial}{\partial r}\left(\lambda r \frac{\partial T}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial}{\partial \theta}\left(\lambda \frac{\partial T}{\partial \theta}\right)+\frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} r1​∂r∂​(λr∂r∂T​)+r21​∂θ∂​(λ∂θ∂T​)+∂z∂​(λ∂z∂T​)+Φ˙=ρCp​∂t∂T​

1r2∂∂r(λr2∂T∂r)+1r2sin⁡2θ∂∂φ(λ∂T∂φ)+1r2sin⁡θ∂∂θ(λsin⁡θ∂T∂θ)+Φ˙=ρCp∂T∂t\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(\lambda r^{2} \frac{\partial T}{\partial r}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial}{\partial \varphi}\left(\lambda \frac{\partial T}{\partial \varphi}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\lambda \sin \theta \frac{\partial T}{\partial \theta}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} r21​∂r∂​(λr2∂r∂T​)+r2sin2θ1​∂φ∂​(λ∂φ∂T​)+r2sinθ1​∂θ∂​(λsinθ∂θ∂T​)+Φ˙=ρCp​∂t∂T​

Thermal Conductivity (导热系数)

λ=∣q∣∣dTdx∣\lambda=\frac{|q|}{\left|\frac{d T}{d x}\right|} λ=∣∣​dxdT​∣∣​∣q∣​

units is W/(m2m^{2}m2K) in the SI system

Thermal Diffusivity (热扩散系数)

a=thermalconductivityheatcapacity=λρCpa=\frac{\text { thermal conductivity }}{\text { heat capacity }}=\frac{\lambda}{\rho C_{p}} a=heatcapacitythermalconductivity​=ρCp​λ​

units is m2/s in the SI system