Question

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.

Answer 1

Answer:

The heat diffusion equation for a 1-D cylindrical, radial coordinate system with internal heat generation is given as

                                           $\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}$

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.

                                             $V_{control}=2\pi rdr$

As per the conservation of energy

                             $\dot{E_{in}}-\dot{E_{out}}+\dot{E_{gen}}=\dot{E_{st}}\\q_r-q_{r+dr}+\dot{q}.V=\rho V c_p\frac{\partial T}{\partial t}$

Now using the Fourier's law in 1-D coordinate system

$q_r=-kA_r \frac{\partial T}{\partial r}\\q_r=-k2 \pi r .1 \frac{\partial T}{\partial r}\\q_r=-2 \pi k r\frac{\partial T}{\partial r}$

Similarly

$q_{r+dr}=q_r+\frac{\partial}{\partial r}(q_r) dr\\q_{r+dr}=-2 \pi k r\frac{\partial T}{\partial r}+\frac{\partial}{\partial r}[-2 \pi k r\frac{\partial T}{\partial r}]dr$

Substituting these values in the energy balance equation gives

$\dot{E_{in}}-\dot{E_{out}}+\dot{E_{gen}}=\dot{E_{st}}\\q_r-q_{r+dr}+\dot{q}.V=\rho V c_p\frac{\partial T}{\partial t}\\-2 \pi k r\frac{\partial T}{\partial r}-[-2 \pi k r\frac{\partial T}{\partial r}+\frac{\partial}{\partial r}[-2 \pi k r\frac{\partial T}{\partial r}]dr]+\dot{q} (2 \pi r dr)=\rho 2 \pi rdr c_p\frac{\partial T}{\partial t}\\-[\frac{\partial}{\partial r}[-2 \pi k r\frac{\partial T}{\partial r}]dr]+\dot{q} (2 \pi r dr)=\rho 2 \pi rdr c_p\frac{\partial T}{\partial t}$

Dividing both sides with 2πr dr

$\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}$

The heat diffusion equation for a 1-D cylindrical, radial coordinate system with internal heat generation is given as

                                           $\frac{1}{\partial r}\frac{\partial}{\partial r}[-k\frac{\partial T}{\partial r}]+\dot{q} =\rho c_p\frac{\partial T}{\partial t}$

This equation is comparable to equation 2.26 when φ and z terms are considered 0.

Answer 2

Answer:

See attachment below

Explanation: