Question$-4$:

a$.$ You are tasked with determining the diameter of a spherical thermocouple junction required to achieve a thermal time constant of one second for measuring the temperature of a gas stream. Relevant parameters include the heat transfer coefficient of $400\frac{W}{m^{2\text{@}}}C,$ thermal conductivity of the thermocouple junction of $20\frac{W}{m}C,$ specific heat capacity of $400\frac{J}{k}gC,$ and density of $8500k\frac{g}{m^3}.$

b$.$ You're also asked to calculate the time it takes for the thermocouple junction to reach $198C$ when initially at $25C$ and placed in a gas stream at $210C.$

a$.$ Sketch the schematic of the system with complete labeling.