A copper wire, 0.8 mm OD, 5 cm long, is placed in an air stream whose temperature rises at a rate given by Tair = (10+ 14t)°C, where t is the time in seconds. If the initial temperature of the wire is 10°C, determine its temperature after 2 s, 10 s and 1 min. The heat transfer coefficient between the air and the wire is 40 W/(m2 K).

Question

GIVEN

• A copper wire is placed in an air stream

• Wire diameter (D) = 0.8 mm=8*10-4 m

• Wire length (L) = 5 cm=0.05 m

• Air stream temperature is: Tair = (10 + 14t)°C

• The initial temperature of the wire (To) = 10°C

• The heat transfer coefficient ( )ch = 40 W/(m2 K)

FIND

• The wire temperature after 2 s, 10 s and 1 min

ASSUMPTIONS

• Constant and uniform heat transfer coefficient

SKETCH



PROPERTIES AND CONSTANTS

For copper at 127°C

• Thermal conductivity (k) = 383 W/(m K)

• Density (?) = 8933 kg/m3

• Specific heat (c) = 383 J/(kg K)

Answer 1

The Biot number for this problem is



Therefore the internal resistance of the wire can be neglected.

The temperature-time history of the wire can be calculated from the energy balance,





This is a linear, first order, non-homogeneous differential equation with a homogeneous solution of T = c e–mt and a particular solution T = co + c1 t. Therefore, the general solution has the form:



Substituting these back into the assumed solution yields





Therefore, the temperature-time history of the wire is



Evaluating the wire temperature at the requested times