Question

Consider a medium in which the heat conduction equation is given in its simplest forms as $$ \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial T}{\partial r}\right)=\frac{1}{\alpha} \frac{\partial T}{\partial t} $$ (a) Is heat transfer steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the thermal conductivity of the medium constant or variable? (e) Is the medium a plane wall, a cylinder, or a sphere? (f) Is this differential equation for heat conduction linear or nonlinear?

Short answer

To summarize, the given heat conduction equation has the following properties: a) Transient heat transfer b) One-dimensional heat transfer c) No heat generation in the medium d) Constant thermal conductivity of the medium e) Cylindrical or spherical shape of the medium (cannot definitively identify) f) Linearity of the differential equation

Step-by-step solution

(a) Steady or Transient Heat Transfer

The given heat conduction equation involves a time derivative term, which is: $$ \frac{\partial T}{\partial t} $$ Since the time derivative term exists in the equation, it means the heat transfer is transient, not steady.

(b) Dimensionality of the Heat Transfer

The equation involves only the spatial variable, r: $$ \frac{\partial}{\partial r}\left(r^{2} \frac{\partial T}{\partial r}\right) $$ As it is the only spatial variable included in the equation, this indicates that the heat transfer is one-dimensional.

(c) Heat Generation in the Medium

The equation does not include any additional terms representing heat generation. Therefore, there is no heat generation in the medium.

(d) Thermal Conductivity of the Medium

The thermal conductivity of the medium is not explicitly given in the equation, but the constant α is used. Since the equation does not involve any function of thermal conductivity as a variable, we can assume that the thermal conductivity is constant.

(e) Shape of the Medium

The equation is given in terms of the radial coordinate, r: $$ \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial T}{\partial r}\right) $$ This suggests that the medium is either cylindrical or spherical in shape. However, without additional information, we can not definitively identify the shape.

(f) Linearity of the Differential Equation

The given differential equation is linear. This can be seen by the absence of any nonlinear interactions with temperature T or its derivatives, such as multiplication of different order derivatives or raising them to a power. The equation only includes the derivatives of the temperature term T and their linear combinations.