Chapter 1, Problem 1.61P

A long bus bar (cylindrical rod used for making electrical connections) of diameter D is installed in a large conduit having a surface temperature of 30°C and in which the ambient air temperature is $T_{\infty }=30°\text{C}$ . The electrical resistivity, ${\rho }_e\left(\mu \Omega \cdot \text{m}\right)$ , of the bar material is a function of temperature, ${\rho }_e={\rho }_e\left[1+\alpha \left(T-T_o\right)\right],$ where ${\rho }_{e,o}=0.0171\mu \Omega \cdot \text{m},T_o=25°\text{C},$ and $\alpha =0.00396{\text{K}}^{\text{-1}}$ . The bar experiences free convection in the ambient air, and the convection coefficient depends on the bar diameter, as well as on the difference between the surface and ambient temperatures. The governing relation is of the form, $h=C{D}^{-0.25}{\left(T-T_{\infty }\right)}^{0.25},$ where $C=1.21\text{W}\cdot {\text{m}}^{-1.75}\cdot {\text{K}}^{\text{-1}}$ . The emissivity of the bar surface is
   $\in =0.85$ .
(a) Recognizing that the electrical resistance per unit length of the bar is ${R^{\prime }}_c={\rho }_e/A_c,$ where $A_c$ is its cross-sectional area, calculate the current-carrying capacity of a 20-mm-diameter bus bar if its temperature is not to exceed 65°C. Compare the relative importance of heat transfer by tree convection and radiation exchange.
(b) To assess the trade-off between current-carrying capacity, operating temperature, and bar diameter. for diameters of 10, 20, and 40 mm. plot the bar temperature T as a function of current for the range $100\le I\le 5000\text{A}$ . Also plot the ratio of the heat transfer by convection to the total heat transfer.