A wire of perimeter P and cross-sectional area A emerges from a die at a temperature T (above ambient temperature) and with a velocity U. Determine the temperature distribution along the wire in the steady state if the exposed length downstream from the die is quite long. State clearly and try to justify all assumptions.

Question

GIVEN

• A wire emerging from a die at a temperature (T) above ambient

• Wire perimeter = P

• Cross-sectional area = A

• Wire emerges at a temperature T above ambient

• Wire velocity = U

FIND

• The temperature distribution along the wire in the steady state if the exposed length downstream from the die is quite long. State clearly and try to justify all assumptions

ASSUMPTIONS

• Ambient temperature is constant at T?

• Heat transfer coefficient between the wire and the air is uniform and constant at hc

• The material properties of the wire are constant

? Thermal conductivity = k

? Thermal diffusivity = ?

• Axial conduction only

• Wire temperature is uniform at a cross section (negligible internal thermal resistance)

SKETCH

Answer 1

Consider a control volume around the wire



Performing an energy balance on the control volume

Conduction into volume + Energy carried into the volume by the moving wire = Conduction out of volume +

Convection to the environment + Energy carried out of the volume by the moving wire.





This is a linear, differential equation with constant coefficients. The solution has the following form



Substituting this solution and its derivatives into the differential equation:



The boundary conditions for the problem are



Applying the first boundary condition



Since, by inspection, s1 must be positive, for the second boundary condition to be satisfied, the constant c1

must be zero. Therefore, the temperature distribution in the wire is