18.9. Civil Engineering Example: Temperature in a Concrete Slab

Try these first

1. 1.

   What is the initial temperature distribution in the concrete slab?

   Hint

   Look for the sine function in the problem statement.

2. 2.

   What are the boundary conditions for this problem?

   Hint

   Consider that the boundaries are insulated.

3. 3.

   What is the initial temperature distribution of the slab?

   Hint

   Refer to the section's description for the initial condition.

4. 4.

   The boundary conditions for the insulation are defined as:

   • A) u(0,t)=u(10,t)=100
   • B) u(0,t)=u(10,t)=0
   • C) u(0,t)≠u(10,t)

   Hint

   Consider what happens at an insulated boundary.

5. 5.

   Consider a concrete slab with modified initial temperature distribution given by u(x,0) = 50 + 20 ext{sin}igg(rac{5 ext{π}x}{10}igg). Determine the steady-state temperature profile as time approaches infinity.

   Hint

   Focus on how oscillating functions behave as they dissipate energy.

6. 6.

   If the thermal diffusivity, α, changes to 0.5, analyze how this affects the decay time of the temperature wave.

   Hint

   Think about how thermal properties influence heat behavior.