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=thermal conductivity heat capacity =λρCpa=\frac{\text { thermal conductivity }}{\text { heat capacity }}=\frac{\lambda}{\rho C_{p}} a= heat capacity thermal conductivity =ρCpλ

units is m2/s in the SI system

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