A long wire of diameter D= 1 mm is submerged in an oil bath of temperature [infinity] T[infinity]= 15°C. The wire has an electrical resistance per unit length of Re′= 0.01 Ω/m. If a current of I= 96 A flows through the wire and the convection coefficient is h= 500 W/m2·K, what is the steady-state temperature of the wire? From the time the current is applied, how long does it take for the wire to reach a temperature that is within 1°C of the steady-state value? The properties of the wire are rho= 8000 kg/m3, c= 500 J/kg·K, and k= 20 W/m·K.

a) what is the steady-state temperature of the wire?

83.67°C

b) From the time the current is applied, how long does it take for the wire to reach a temperature that is within 1°C of the steady-state value?

8.31s

Explanation:

a) what is the steady-state temperature of the wire?

From the above question we are given the following values:

Oil bath temperature = 15°C.

Diameter of wire = 1mm

Converting to m(meter)

1 mm = 0.001m

The wire has an electrical resistance per unit length of Re′= 0.01 Ω/m. Current of I= 96 A

Convection coefficient is h= 500 W/m²·K,

Temperature in steady state formula =

Temperature + I² × Re' /π × D × h

= 25°C + 96² × 0.01/π × 0.001 × 500

= 25°C + 92.16/1.5707963268

= 25°C + 58.670878232

= 83.670878232°C

≈ 83.67°C

b)From the time the current is applied, how long does it take for the wire to reach a temperature that is within 1°C of the steady-state value?

For question b, we are given the following values

rho= 8000 kg/m3, c= 500 J/kg·K, and k= 20 W/m·K

Formula

= Exp( -4 × h/rho × c × k)

=Exp(- 4 × 500 W/m²·K)/8000 kg/m³ ×

500 J/kg·K × 20 W/m·K

= 8.31s