Heat Transfer Problem :
Maximum current in an electric wire
Problem.
An electric wire with radius r0 of 0.5 mm is made of copper [electrical conductivity = 5.1 x 107 ohm-1 m-1 and thermal conductivity = 380 W/(m K)]. It is insulated (see figure) to an outer radius r1 of 1.5 mm with plastic [thermal conductivity = 0.35 W/(m K)].
Figure. Heating of an insulated electric wire.
The ambient air is at 38oC and the heat transfer coefficient from the outer insulated surface to the surrounding air is 8.5 W/(m2 K). Determine the maximum current in amperes that can flow at steady-state in the wire without any portion of the insulation getting heated above its maximum allowable temperature of 93oC.
Solution.
Step. Thermal resistance representation for insulation and air
In general, the heat flow is given by Q = T/Rth,
where T is the temperature driving force (thermal potential difference).
The thermal resistance for a cylindrical annulus is Rth = ln (r1/r0)/(2kL) and the thermal resistance for a fluid film at a solid-fluid interface is Rth = 1/(hA). Here, k is the thermal conductivity, h is the heat transfer coefficient and A is the surface area for convection.
The thermal resistances for the insulation and air film are in series as shown in the figure below.
Figure. Thermal resistance representation of insulation and air film.
Based on the above thermal resistance representation, the heat flow is
(1)
where k is the thermal conductivity of the plastic insulation.
Step. Heat flow due to current in wire
The flow of an electric current results in some electrical energy getting converted to thermal energy irreversibly. The heat generation by electrical dissipation per unit volume is given by S = I2/ke where I is the current density (in amp/m2) and ke is the electrical conductivity (in ohm-1 m-1).
The total heat generated within the wire is simply the product of S and the volume of the wire. At steady-state, all this heat generated within the wire by electrical dissipation must leave through the wire surface and therefore the heat flow is given by
(2)
Step. Expression for current
On eliminating Q from the above two equations, the current density is
(3)
On multiplying the current density by the cross-sectional area of the wire, the current is obtained from
(4)
For the maximum current, the temperature T0 must be maximized.
Step. Substitution of numerical values
On setting the temperature T0 to 93oC (i.e., the maximum allowable temperature for the insulation), substitution of the numerical values gives the maximum current that can flow through the wire as
Step. Thermal resistance representation for insulation and air
T/Rth,
where 
kL) and the thermal resistance for a fluid film at a solid-fluid interface is Rth = 1/(hA). Here, k is the thermal conductivity, h is the heat transfer coefficient and A is the surface area for convection.
![equation : Q = (T_0 - T_2)/R_th where R_th = ln (r_1/r_0)/2 pi k L + 1 / (h 2 pi r_1 L]](/iosundry/heat/heat1006/img1.gif){kind=small}
