Question

A 1 m long wire of diameter 1mm is submerged in an oil bath of temperature 25-degC. The wire has an electrical resistance per unit length of 0.01 Ω/m. If a current of 100 A flows through the wire and the convection coefficient is 500W/m2K, 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 within 1-degC of the steady state value? The density of the wire is 8,000kg/m3, its heat capacity is 500 J/kgK and its thermal condu

Answer 1

steady state temperature =88.7deg C

t=time within 1 deg C of it steady state is 8.31s

Step-by-step explanation:

A 1 m long wire of diameter 1mm is submerged in an oil bath of temperature 25-degC. The wire has an electrical resistance per unit length of 0.01 Ω/m. If a current of 100 A flows through the wire and the convection coefficient is 500W/m2K, 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 within 1-degC of the steady state value? The density of the wire is 8,000kg/m3, its heat capacity is 500 J/kgK and its thermal condu

The diameter of the wire is known to be=1mm

properties=

The density of the wire is 8,000 kg/m3,

heat capacity is 500 J/kgK

themal conductivity is 20W/m.K

electrical resistance per unit length of 0.01 Ω/m

from lump capavity method

`B_(i) =(hr/2)/(k)`

500*(2.5*10^-4)/20

0.006<0.1

we know also, to find steady state temperature

`\pi`Dh(T-Tinf)=
`I^(2) R_(e)`

make T the subject of the equation , we have

T=25+
`(100^2*0.01)/(\pi*0.001*500 )`

T=88.7 degC

rate of chnage in temperature

dT/dt=
`(I^2*Re)/(rho*c*\pi*D^2/4 ) -(4h)/(rho*c*D) (T-Tinf)`

at t=o and integrating both sides
`(T-Tinf-(I^2*Re/\pi*Dh) )/(Ti-Tinf-(I^2*Re/\pi*Dh ) =exp(-4ht)/(rho*c*D)`

we have

`(87.7-25-63.7)/(25-25-63.7) =exp(4*500t)/(8000*500*0.001)`

t=8.31s

steady state temperature =88.7deg C

t=time within 1 degC of it steady stae is 8.31s