4. A thermometer initially at is immersed in a vacuum furnace whose walls are at C. Formulate the thermometer response as a first order system and determine its time constant. Assume black body radiation. Assume the bead-to-furnace shape factor to be unity. Stefan-Boltzmann constant W/m² K ⁴; mass of the sensor g, specific heat and surface area . Hint : Show that the thermocouple bead temperature will increase as per the differential equation

where is the mass of the bead, is its surface area. The solution of the differential equation can be obtained analytically, by linearizing the second term; alternatively, an iterative numerical solution can be obtained, say, by a finite difference scheme.