Physics, 19.02.2020 05:58 modelingavakin
2.44 Beginning with a differential control volume in the form of a cylindrical shell, derive the heat diffusion equation for a one-dimensional, cylindrical, radial coordinate system with internal heat generation. Compare your result with Equation 2.26.
Answers: 2
![\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}\\](/tpl/images/0515/5047/abaa5.png)
![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](/tpl/images/0515/5047/7bdbf.png)
![\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}\\](/tpl/images/0515/5047/f1d36.png)
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2.44 Beginning with a differential control volume in the form of a cylindrical shell, derive the hea...
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