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/m3, 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 Tf is 120°C and the initial temperature of the sphere T0 is 25°C. The heat transfer coefficient between the gas and the junction is h = 70 W/(m2・°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 Tf.
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.