Chapter 3, Problem 3.156P

Small spherical particles of diameter $D=50\mu m$ contain a fluorescent material that, when irradiated with white light, emits at a wavelength corresponding to the material’s 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 how by observing their emitted color. If the particles arc characterized by a density, specific heat, and thermal conductivity of $\rho =999kg/m^3,k=1.2W/m\cdot K,\text{and}{c}_p=1200J/kg\cdot K$, 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.