In a metal manufacturing plant and its heat treatment process, a long, hexagonal copper extrusion (or rod) comes out of heat-treatment furnace at 400°C and is then quenched by immersing it in a 50°C air stream flowing perpendicular to its axis at 10 m/s as depicted in the figure. The surface of the copper has an emissivity of 0.9 due to oxidation in this process. The rod is 3 cm across opposing flat sides, and it has a cross-sectional area of 7.79 cm2, and a perimeter of 10.4 cm. Determine the time required for the center of the copper to cool to 100°C.

Question

GIVEN

• A long hexagonal copper extrusion in an air stream flowing perpendicular to its axis

• Initial temperature (To) = 400°C

• Air temperature (T?) = 50°C

• Air velocity (V?) = 10 m/s

• Surface emissivity (?) = 0.9

• Distance across the flats (D) = 3 cm = 0.03 m

• Cross sectional area of the extrusion (Ac) = 7.79 cm2 = 7.79 × 10–4 m2

• Perimeter of the extrusion (P) = 10.4 cm = 0.104 m

FIND

• The time (t) required for the center of the copper to cool to 100°C

ASSUMPTIONS

• Variations of the copper properties with temperature are negligible

SKETCH



PROPERTIES AND CONSTANTS

Thermal conductivity (ka) = 0.0339 W/(m K) Kinematic viscosity (?) = 29.6 × 10–6 m2/s Prandtl number (Pr) = 0.71 For Appendix 2, Table 12, for copper

Thermal Conductivity (k) = 386 W/(m K) at 250°C Density (?) = 8933 kg/m3 at 20°C Specific heat (c) = 383 J/(kg K) at 20°C

Answer 1

The Reynolds number is



The Nusselt number for non-circular cross sections in gases by



where D, B, and n are given B = 0.138, n = 0.638



The characteristic length for determining the Biot number of the rod is defined in



The Biot Number, is



Therefore, the internal thermal resistance of the extrusion may be neglected and lumped parameters

may be applied. An energy balance on the extrusion, including radiation, yields the following



This equation must be solved numerically



This can be solved numerically using a finite difference method