CATALOG

1D steady conduction problem without heat souce:a) plane wallsb) pipe wallsc) shells

1D steady conduction problem without heat souce:

The totle gavorning function

1rn∂∂r(λrn∂T∂r)+Φ˙=ρCp∂T∂t\frac{1}{r^{n}} \frac{\partial}{\partial r}\left(\lambda r^{n} \frac{\partial T}{\partial r}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t}rn1​∂r∂​(λrn∂r∂T​)+Φ˙=ρCp​∂t∂T​

n=0foraplanewalln=1foracylindern=2forasphere\mathrm{n}=0 \quad \text{for a plane wall}\\ \mathrm{n}=1 \quad \text{for a cylinder}\\ \mathrm{n}=2 \quad \text{for a sphere} n=0foraplanewalln=1foracylindern=2forasphere

Cartesian Coordinate

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

Cylindrical Coordinate

1r∂∂r(λr∂T∂r)+Φ˙=ρCp∂T∂tintherdirection\frac{1}{r} \frac{\partial}{\partial r}\left(\lambda r \frac{\partial T}{\partial r}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} \quad\text { in the } r \text { direction } r1​∂r∂​(λr∂r∂T​)+Φ˙=ρCp​∂t∂T​intherdirection

Spherical Coordinate

1r2∂∂r(λr2∂T∂r)+Φ˙=ρCp∂T∂tintherdirection\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(\lambda r^{2} \frac{\partial T}{\partial r}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} \quad\text { in the } r \text { direction } r21​∂r∂​(λr2∂r∂T​)+Φ˙=ρCp​∂t∂T​intherdirection

a) plane walls

R=δ1λ1+δ2λ2+δ3λ3R=\frac{\delta_{1}}{\lambda_{1}}+\frac{\delta_{2}}{\lambda_{2}}+\frac{\delta_{3}}{\lambda_{3}} R=λ1​δ1​​+λ2​δ2​​+λ3​δ3​​

ϕ=Tw1−Tw4A(δ1λ1+δ2λ2+δ3λ3)\phi=\frac{T_{w 1}-T_{w 4}}{A(\frac{\delta_{1}}{\lambda_{1}}+\frac{\delta_{2}}{\lambda_{2}}+\frac{\delta_{3}}{\lambda_{3}})} ϕ=A(λ1​δ1​​+λ2​δ2​​+λ3​δ3​​)Tw1​−Tw4​​

b) pipe walls

Rt=R1+R2+R3=ln⁡(r2/r1)2πLλ1+ln⁡(r3/r2)2πLλ2+ln⁡(r4/r3)2πLλ3\begin{aligned} R_{t} &=R_{1}+R_{2}+R_{3} \\ &=\frac{\ln \left(r_{2} / r_{1}\right)}{2 \pi L \lambda_{1}}+\frac{\ln \left(r_{3} / r_{2}\right)}{2 \pi L \lambda_{2}}+\frac{\ln \left(r_{4} / r_{3}\right)}{2 \pi L \lambda_{3}} \end{aligned} Rt​​=R1​+R2​+R3​=2πLλ1​ln(r2​/r1​)​+2πLλ2​ln(r3​/r2​)​+2πLλ3​ln(r4​/r3​)​​

Φ=2πL(T1−T4)ln⁡(r2/r1)λ1+ln⁡(r3/r2)λ2+ln⁡(r4/r3)λ3\Phi=\frac{2 \pi L\left(T_{1}-T_{4}\right)}{\frac{\ln \left(r_{2} / r_{1}\right)}{\lambda_{1}}+\frac{\ln \left(r_{3} / r_{2}\right)}{\lambda_{2}}+\frac{\ln \left(r_{4} / r_{3}\right)}{\lambda_{3}}} Φ=λ1​ln(r2​/r1​)​+λ2​ln(r3​/r2​)​+λ3​ln(r4​/r3​)​2πL(T1​−T4​)​

c) shells

one shell:

R=14πλ(1r1−1r2)R=\frac{1}{4 \pi \lambda}\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right) R=4πλ1​(r1​1​−r2​1​)