Final answer:
The centerline temperature of the rod in forced convection can be approximated as the average of the furnace and surface temperatures. Assuming a linear temperature gradient, the centerline temperature would be 650 K.
Step-by-step explanation:
The question asks about the centerline temperature of a rod when its surface temperature is at a given value. This situation involves concepts from heat transfer, specifically forced convection and steady-state heat conduction in the rod. The key parameters given are the rod's diameter (0.06 m), density (ρ = 8000 kg/m3), specific heat (c = 500 J/kg·K), and thermal conductivity (k = 50 W/m·K). The convection coefficient is 1000 W/m2·K, and the furnace temperature is 750 K. To find the centerline temperature, generally, a detailed solution requires solving the heat conduction differential equation considering the convection boundary condition at the surface.
However, this problem can be considerably simplified if we assume that the temperature gradient within the rod is linear between the center and surface. Under this assumption, which can be valid for a long, thin rod with good thermal conductivity, the centerline temperature would be the average of the furnace temperature and the surface temperature of the rod. Mathematically, T_centerline = (T_furnace + T_surface) / 2. With the values provided, we would get: T_centerline = (750 K + 550 K) / 2 = 650 K.
Do note that this simple linear gradient approach is an approximation and the actual situation might require more sophisticated analysis through the use of partial differential equations and numerical methods if a more precise solution is necessary.