Develop a one-dimensional transient numerical solution for a plane wall of a nuclear containment vessel. The external wall is held at a constant temperature of 21 degrees C. The 30 cm wall is made of concrete and is exposed to a heat flux on the interior side. At normal steady state operation, the reactor subjects the interior of the wall to q''_op = 1.2 kW/(m^2). The NRC has asked you to evaluate the safety of the container if the reactor were to enter a meltdown scenario. During meltdown, the flux increases exponentially in time such that q''= q''_op*exp(t/1000), where t is the time is seconds after the initiation of the meltdown process. If the concrete can withstand 1100 degrees C, how long do plant operators have before the container fails?
In answering the problem, provide the following analysis:
1. Verify your finite difference code solution against a semi-infinite slab solution. For this verification, use a constant surface heat flux boundary condition (q''_op = 1.2 W/m2) instead of the meltdown condition.
2. Explain how you established the appropriate Fourier number (i.e. grid independence and time step)
3. Modify the finite difference MATLAB code to include the meltdown surface heat flux (time dependent). Plot the temperature distribution at t = 0 and two other times prior to container failure. Assume the initial temperature is the average between the steadystate interior and exterior temperatures of the concrete.