Question 13.1: A cylindrical thermometer element, diameter 3 mm is enclosed...
A cylindrical thermometer element, diameter 3 mm is enclosed in a radiation shield that excludes all solar radiation but is 5.0 °C warmer than the true air temperature which is 20 °C. If the thermometer is to record a temperature within 0.1 °C of true air temperature, at what windspeed must the element be ventilated? (Assume long-wave emissivities of 1.0.)
The resistance of the thermometer to sensible heat transfer is r_{H} = {d}/{kNu} . The Nusselt number is given by Nu=0.24Re^{0.60} = 0.24 \times (3\times 10^{-3} )^{0.6} \times V^{0.6} = 5.65 \; V^{0.6} . Hence r_{H} = 3\times 10^{-3} /(22.2\times 10^{-6} \times 5.65\times V^{0.6} ) =23.9\; V^{-0.6} \; s \; m^{-1} . The radiative resistance to heat transfer is r_{R} = \rho c_{p} / 4\sigma T^{3} = 210\; s \; m^{-1} . From Eq. 13.5,
T_{t} = (r_{H} T_{s} + r_{R} T ) /(r_{R} + r_{H} ) (13.5)
T_{t} = (r_{H} T_{sh} + r_{R} T_{a} ) /(r_{R} + r_{H} ) . Rearranging, r_{H} = r_{R} (T_{t} -T_{a} )/(T_{sh} -T_{t} ) , which must be 0.1 × 210/4.9 = 4.3 s m^{−1}. Hence, V ^{−0.6} = 4.3/23.9, and V = 17.6 m s^{−1}.