Answer :
Answer:
The heat diffusion equation for a 1-D cylindrical, radial coordinate system with internal heat generation is given as
[tex]\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}\\[/tex]
This equation is comparable to equation 2.26 when φ and z terms are considered 0.
Explanation:
As per the given statement, heat diffusion equation for a one-dimensional, cylindrical, radial coordinate system with internal heat generation is given as the condition.
Let us consider a control volume of unit thickness with volume as given below perpendicular to the paper.
[tex]V_{control}=2\pi rdr[/tex]
As per the conservation of energy
[tex]\dot{E_{in}}-\dot{E_{out}}+\dot{E_{gen}}=\dot{E_{st}}\\q_r-q_{r+dr}+\dot{q}.V=\rho V c_p\frac{\partial T}{\partial t}[/tex]
Now using the Fourier's law in 1-D coordinate system
[tex]q_r=-kA_r \frac{\partial T}{\partial r}\\q_r=-k2 \pi r .1 \frac{\partial T}{\partial r}\\q_r=-2 \pi k r\frac{\partial T}{\partial r}[/tex]
Similarly
[tex]q_{r+dr}=q_r+\frac{\partial}{\partial r}(q_r) dr\\q_{r+dr}=-2 \pi k r\frac{\partial T}{\partial r}+\frac{\partial}{\partial r}[-2 \pi k r\frac{\partial T}{\partial r}]dr[/tex]
Substituting these values in the energy balance equation gives
[tex]\dot{E_{in}}-\dot{E_{out}}+\dot{E_{gen}}=\dot{E_{st}}\\q_r-q_{r+dr}+\dot{q}.V=\rho V c_p\frac{\partial T}{\partial t}\\-2 \pi k r\frac{\partial T}{\partial r}-[-2 \pi k r\frac{\partial T}{\partial r}+\frac{\partial}{\partial r}[-2 \pi k r\frac{\partial T}{\partial r}]dr]+\dot{q} (2 \pi r dr)=\rho 2 \pi rdr c_p\frac{\partial T}{\partial t}\\-[\frac{\partial}{\partial r}[-2 \pi k r\frac{\partial T}{\partial r}]dr]+\dot{q} (2 \pi r dr)=\rho 2 \pi rdr c_p\frac{\partial T}{\partial t}\\[/tex]
Dividing both sides with 2πr dr
[tex]\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}\\[/tex]
The heat diffusion equation for a 1-D cylindrical, radial coordinate system with internal heat generation is given as
[tex]\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}\\[/tex]
This equation is comparable to equation 2.26 when φ and z terms are considered 0.
