Some advanced combustion systems operate in a pulsed mode where an acoustic wave resonates in the combustor to enhance heat and mass transfer (called pulsed combustors, concept initially used in World War II in infamous Buzz Bombs more recently used by Lennox in high efficiency furnaces). The frequency of the wave is approximately 100Hz (cycles/sec). As a result, the temperature at any given spatial location in the combustor fluctuates with time. For our purposes, the fluctuations can be approximated as a sine wave with an amplitude of 150C for the system of interest. We would like to measure the fluctuating temperature at a given point in the reactor using a thermocouple. To do this, we need a dynamic relationship between the temperature measured by the thermocouple (T) and the fluctuating gas temperature Tg. Heat is transferred to the thermocouple by convection as expected. Heat transfer to the thermocouple by radiation is also important and is approximated by the following expression: q' = e(Tftame T4) where & is the emissivity and O is the Stefan- Boltzmann constant (W/(mK4). The thermocouple bead itself can be approximated as a sphere (e.g. ignore conduction along the connecting wires, etc). a) Write a dynamic model that can be solved for the thermocouple bead temperature (T.) as a function of time. b) Linearize the model about a set of steady-state temperatures Tg, TFlame, and T, and transform the equation using derivative variables Note: Results from part b will be used in SP 10