10.6.1. Application: Heat Equation in a Semi-Infinite Rod

Try these first

1. 1.

   Define the heat equation for a semi-infinite rod.

   Hint

   Think about how temperature changes over time.

2. 2.

   What constant temperature is held at x=0?

   Hint

   This is what you will observe at one end of the rod.

3. 3.

   What is the general form of the heat equation for a semi-infinite rod?

   • a) \\( \\frac{\\partial u}{\\partial t} = \\alpha^2 \\frac{\\partial^2 u}{\\partial x^2} \\)
   • b) \\( \\frac{\\partial u}{\\partial x} = \\alpha^2 \\frac{\\partial^2 u}{\\partial t^2} \\)
   • c) \\( \\frac{\\partial u}{\\partial t^2} = \\alpha^2 \\frac{\\partial^2 u}{\\partial x^2} \\)

   Hint

   Look for how temperature changes over time in this differential equation.

4. 4.

   True or False: The Fourier Cosine Transform is only applicable to bounded domains.

   • True
   • False

   Hint

   Consider where you encounter cosine transformations in boundary value problems.

5. 5.

   In a semi-infinite rod made of copper, calculate the time taken for the temperature to change significantly (by 50%) at a distance of 1 meter from the heated end if the thermal diffusivity is 1.1 × 10^{-4} m²/s.

   Hint

   Refer to the error function for determining significant temperature changes.

6. 6.

   Discuss how altering the boundary condition to a time-varying temperature at one end of the rod would impact the solution structure. What complications arise?

   Hint

   Think about how time-variability influences the interpretation of heat flow.