The junction of a thermocouple can be approximated as a sphere with a diameter of 1 mm. As shown in Figure 7.32, the thermocouple is used to measure the temperature of a gas stream. For the junction, the density is ρ = 8500 kg/m^{3}, the specific heat capacity is

*c *= 320 J/(kg・°C), and the thermal conductivity is *k *= 40 W/(m・°C). The temperature of the gas *T*_{f} is 120°C and the initial temperature of the sphere *T*0 is 25°C. The heat transfer coefficient between the gas and the junction is *h *= 70 W/(m^{2}・°C).

a. Determine if the junction’s temperature can be considered uniform.

b. Derive the differential equation relating the junction’s temperature *T*(*t*) and the gas’s temperature *T*_{f}.

c. Using the differential equation obtained in Part (b), construct a Simulink block diagram to find out how long it will take the thermocouple to read 99% of the initial temperature difference.