Continuous temperature monitoring in exothermic reactors is critical for safety. The temperature profile across reaction zones is modeled by piecewise functions, which must be continuous and differentiable to avoid physical discontinuities in heat transfer models.
An exothermic chemical reactor has a temperature profile along its central axis modeled by the piecewise function:
where is the distance in meters from the inlet, and is the temperature in degrees Celsius.
Determine the following:
- Find the values of and such that the temperature profile is continuous and differentiable at the boundary meters.
- Optimize the reactor design by finding the maximum temperature along the axis in the interval .