if the extrusion cross-section is elliptical with the major axis normal to the air flow and same mass per unit length. The major axis of the elliptical cross-section is 5.46 cm and its perimeter is 12.8 cm

Question

GIVEN

• A long elliptical copper extrusion in an air stream

• Initial temperature (To) = 400°C

• Air Temperature (T?) = 50°C

• Air velocity (V?) = 10 m/s

• Surface emissivity (?) = 0.9

• Elliptical cross-section with major axis normal to the air flow

• Length of the major axis of the ellipse (D) = 5.46 cm = 0.0546 m

• Perimeter of ellipse (P) = 12.8 cm = 0.128 m

• Same mass per unit length as Problem 6.9

FIND

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

ASSUMPTIONS

• Air flow is perpendicular to the axis of the extrusion

• Variation of the copper properties with temperature is negligible

SKETCH



PROPERTIES AND CONSTANTS

the initial and final film temperature of 150°C

Thermal conductivity (ka) = 0.0339 W/(m K) Kinematic viscosity (?) = 29.6 × 10–69 m2/s Prandtl number (Pr) = 0.71

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

Since the density of the extrusion in this problem is the same as the previous problem, the same mass

per unit length implies the same cross-section area



Following the same procedure as the solution

The Reynolds number is



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



where D, B, and n are given by B = 0.085, n = 0.804

(Although the Reynolds number for this case is slightly out of range it will be

applied to estimate the Nusselt number)



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



The Biot number, from as



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 can be solved numerically using a finite difference method



COMMENTS

The elliptical extrusion cools more quickly due to both higher convection heat transfer coefficient and

more surface area.