Question

A long plastic rod of 20-mm diameter (k = 0.3 W/m · K and rhocp = 1040 kJ/m3 · K) is uniformly heated in an oven as preparation for a pressing operation. For best results, the temperature in the rod should not be less than 200°C. To what uniform temperature should the rod be heated in the oven if, for the worst case, the rod sits on a conveyor for 3 min while exposed to convection cooling with ambient air at 25°C and with a convection coefficient of 10 W/m2 · K? A further condition for good results is a maximum–minimum temperature difference of less than 10°C. Is this condition satisfied? If not, what could you do to satisfy it?

Answer 1

Answer:

The temperature that would satisfy is 286.7°C

Explanation:

Given data:

d = 20 mm = 0.02 m

Le = characteristic length = 0.02/4 = 5x10⁻³m

K = 0.3 W/m K

ρCp = 1040 kJ/m³ K = 1.04x10⁶

t = 3 min = 180 s

h = 10 W/m² K

According transient heat analysis:

$\frac{T-25}{T_{o}-25 } =e^{-(\frac{h}{L_{e}\rho Cp } )t}$

$\frac{200-25}{T_{o} -25} =e^{-(\frac{10}{5x10^{-3}*1.04x10^{6} })*180 } \\T_{o} =272.38C$

200°C is the limiting temperature, the temperature should not be less to 200°C, then, we suppose a temperature of 210°C (10°C), and find the temperature:

$\frac{210-25}{T_{o} -25} =e^{-(\frac{10}{5x10^{-3}*1.04x10^{6} })*180 } \\T_{o} =286.7C$