Consider a rotating disk suspended in an infinite fluid. One remarkable result for laminar flow situations was that the boundary layer thickness and hence transport coefficients were uniform across the surface of the disk. Using the 1/7th power law velocity profile, it has been shown that the boundary layer thickness in turbulent flow is a function of radial position:

\[\delta_{h}=0.526 r\left(\frac{v}{\omega r^{2}}\right)^{1 / 5} \quad \operatorname{Re}=\frac{\omega r_{o}^{2}}{v}\]

The analog of the friction factor, the average torque coefficient, \(\overline{C_{T}}\), is defined by:

\[\overline{C_{T}}=\frac{2 T}{\frac{1}{2} ho \omega^{2} r_{o}^{5}}\]

where \(r_{o}\) is the radius of the disk and \(\omega\) is the rotation rate. In turbulent flow:
\[\begin{gathered}\overline{C_{T}}=0.146 \operatorname{Re}^{-1 / 5} \quad(1 / 7 \text { th power law) } \\\frac{1}{\sqrt{\overline{C_{T}}}}=1.97 \log _{10}\left(\operatorname{Re} \sqrt{\overline{C_{T}}}\right)+0.03 \quad \text { (Universal profile) }\end{gathered}\]

a. For a disk \(10 \mathrm{~cm}\) in diameter, immersed in water at \(25^{\circ} \mathrm{C}\), and spinning at \(3000 \mathrm{rpm}\), what is the boundary layer thickness at the edge of the plate, and what is the total torque required to spin the disk?

b. The shear stress at the surface of the disk is proportional to the boundary layer thickness, i.e.:
\[\tau_{w} \approx ho r \omega^{2} \delta_{h}\]
Using the Chilton-Colburn analogy, determine the functional form of a correlation for the heat and mass transfer coefficient in turbulent flow.