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).
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)
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
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