Chapter 5, Problem 5.17P

Small spherical particles of diameter $D=50\mu \text{m}$ contain a fluorescent material that. when irradiated with white light, emits at a wavelength corresponding to the materials temperature. Hence the color of the particle varies with its temperature. Because the small particles are neutrally buoyant in liquid water. a researcher wishes to use them to measure instantaneous local water temperatures in a turbulent flow by observing their emitted color. If the particles are characterized by a density. specific heat, and thermal conductivity of $\rho =999{\text{kg/m}}^{\text{3}},k=1.2\text{W/m}\cdot \text{K,}$ and $c_p=1200\text{J/kg}\cdot \text{K,}$ respectively, determine the time constant of the particles. Hint: Since the particles travel with the flow. heat transfer between the particle and the fluid occurs by conduction. Assume lumped capacitance behavior.