A spherical thermometer junction of diameter 0.706 mm is to be used for the measurement of temperature of a gas stream. The convective heat transfer co-efficient on the bead surface is 400 W/m^2K. Thermo-physical properties of thermocouple material are k = 20W/mK, C = 400 J/kg K and \rho = 8500 kg/m³. If the thermocouple initially at 30°C is placed in a hot stream of 300°C, the time taken by the bead to reach 298°C is
- A. 2.35 s
- B. 4.9 s
- C. 14.7 s
- D. 29.4 s
Answer: B. 4.9 s
Explanation: Lumped capacitance: Bi = h(r/3)/k. r=0.353mm. Bi < 0.1, valid. Time constant \tau = \rho V C / h A = \rho (r/3) C / h. T = \rho d C / 6h = 8500*0.000706*400 / (6*400) = 1.0 s. (T-Tinf)/(Ti-Tinf) = e^(-t/\tau). (298-300)/(30-300) = -2/-270 = 0.0074. ln(0.0074) = -t/1. -4.9 = -t. t=4.9s.