consider a solid stainless steel wire with a thermal con- ductivity of 14 w/m⋅k. the wire has a diameter of 1 mm, has a resistivity of 45 × 10−8ω⋅m, and carries a current of 120 a. (a) determine the rate of heat generated within the wire (w/m3 ), and (b) calculate the maximum temperature rise in the wire.

Heat is the energy that has a correlation with the resistivity of the material. Resistivity depends on the resistance and wire dimension where the resistance of the material can be written as

R = ρ.L/A

where R is resistance, ρ is resistivity, L is the length of the wire and A is surface area.

The given parameters from this question are

σ = 14 W/m.K

ρ = 45 x 10¯⁸ Ωm

I = 120 A

d = 1 mm

r = 0.5 mm = 5 x 10¯⁴ m

Calculate the surface area

A = πr²

A = π(5 x 10¯⁴)²

A = 7.86 x 10¯⁷ m²

In 1 m³, the length of wire should be

V = A . L

1 = 7.86 x 10¯⁷ . L

L = 1272727.273 m

Find the resistance of wire per m³

R/V = ρ.L/A

R/V = 45 x 10¯⁸ . 1272727.273 / 7.86 x 10¯⁷

R/V = 728925.62 Ω/m³

Find the heat generated per m³ by using power

P/V = I² . R

P/V = 120² . 728925.62

P/V = 1.05 x 10¹⁰ W/m³

Find the heat generated per m by using power

P/L = P/V

P/L = P/V . A

P/L = 1.05 x 10¹⁰ . 7.86 x 10¯⁷

P/L = 8253 W/m

Find the maximum temperature

T = (P/L) / σ

T = 8253 / 14

T = 589.5 K

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