Thermocouples are devices used to measure temperature of a given sample or the surrounding medium. These devices feature a “bead”, which has a spherical shape, and produces voltage upon change in...

Thermocouples are devices used to measure temperature of a given sample or the surrounding
medium. These devices feature a “bead”, which has a spherical shape, and produces voltage upon change
in temperature. For an engineering application at a pharmaceutical company, thermocouple devices are
being tested for their responsiveness, i.e. how fast it can detect temperature changes in the environment
(surrounding air). The engineers at this company have specified design constraints for an ideal
thermocouple: it must detect temperature changes no later than 1.5 minute, and the reported
temperature value must be reasonably correct: at most %3 difference between measured and actual
temperature values is allowed.
Four thermocouples from different vendors are being tested. Relevant properties of these devices are
listed below:

The experiment involves placing the thermocouple from air at 20 °C to air at 150 °C, and monitoring the
time it takes for the measured values to reach 150 °C, which is the temperature value the thermocouple
should read in this case. If the convection heat transfer coefficient of the air is 10 W/m2*K, which
thermocouple meets the design criteria the best? Please perform necessary calculations.

Extracted text: Device Thermal conductivity Density Specific Heat Bead Diameter Thermocouple 1 36 W/m*K 12000 kg/m3 10000 kg/m3 12000 kg/m3 9000 kg/m3 140 kJ/kg*K 120 kJ/kg*K 150 kJ/kg*K 160 kJ/kg*K 0.9 mm Thermocouple 2 36 W/m*K 0.9 mm Thermocouple 3 Thermocouple 4 42 W/m*K 1.2 mm 33 W/m*K 0.7 mm