The addition of insulation to a cylindrical surface, such as a wire, may increase the rate of heat dissipation to the surroundings (a) For a No. 10 wire (0.26 cm in diameter), what is the thickness of rubber insulation [k = 0.16 W/(m K)] that maximizes the rate of heat loss if the heat transfer coefficient is 10 W/(m2 K)? (b) If the current-carrying capacity of this wire is considered to be limited by the insulation temperature, what per cent increase in capacity is realized by addition of the insulation? State your assumptions.

Question

GIVEN

• An insulated cylindrical wire

• Diameter of wire (Dw) = 0.26 cm = 0.0026 m

• Thermal conductivity of rubber (k) = 0.16 W/(m K)

• Heat transfer coefficient ( h c) = 10 W/(m2 K)

FIND

(a) Thickness of insulation (Li) to maximize heat loss (b) Per cent increase in current carrying capacity

ASSUMPTIONS

• The system is in steady state

• The thermal conductivity of the rubber does not vary with temperature

SKETCH

Answer 1

(a) the radius that will maximize the rate of heat transfer (rc) is:



The thickness of insulation needed to make this radius is



(b) The thermal circuit for the insulated wire is shown below



The rate of heat transfer from the wire is



If only a very thin coat of insulation is put on the wire to insulate it electrically then ro = ri = Dw/2 = 0.0013 m. The rate of heat transfer from the wire is



For the wire with the critical insulation thickness



The current carrying capacity of the wire is directly related to the rate of heat transfer from the wire. For a given maximum allowable insulation temperature, the increase in current carrying capacity of the wire with the critical thickness of insulation over that of the wire with a very thin coating of insulation is