Chapter 5, Problem 5.69P

The density and specific heat of a particular material are known $\left(\rho =1200{\text{kg/m}}^{\text{3}},c_p=1250\text{J/kg}\cdot \text{K}\right),$ but its thermal conductivity is unknown. To determine the thermal conductivity, a long cylindrical specimen of diameter $D=40\text{mm}$ is machined, and a thermocouple is inserted through a small hole drilled along the centerline.

The thermal conductivity is determined by performing an experiment in which the specimen is heated to a uniform temperature of $T_i=100°\text{C}$ and then cooled by passing air at $T_{\infty }=25°\text{C}$ in cross flow over the cylinder. For the prescribed air velocity, the convection coefficient is $h=55{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$

1. If a centerline temperature of $T\left(0,t\right)=40°\text{C}$ is recorded after $t=1136\text{s}$ of cooling, verify that the material has a thermal conductivity of $k=0.30\text{W/m}\cdot \text{K}\text{.}$
2. For air in cross flow over the cylinder, the prescribed value of $h=55{\text{W/m}}^{\text{2}}\cdot \text{K}$ corresponds to a velocity of $V=6.8\text{m/s}\text{.}$ If $h=C{V}^{0.618},$ where the constant C has units of $\text{W}\cdot {\text{s}}^{0.618}{\text{/m}}^{2.618}\cdot \text{K,}$ how does the centerline temperature at $t=1136\text{s}$ vary with velocity for $\text{3}\le \text{V}\le \text{20}\text{m/s?}$ Determine the centerlinetemperature histories for $0\le t\le 1500\text{s}$ and velocities of 3, 10, and 20 m/s.