How long should a nichrome wire with a cross section of 0.5 mm2 be taken to make an electric heater

How long should a nichrome wire with a cross section of 0.5 mm2 be taken to make an electric heater from it, with which you can completely melt 1 kg of ice at 0 degrees in 10 minutes? Mains voltage 220 V, heater efficiency 60%.

S = 0.5 mm2.

ρ = 1.1 Ohm * mm2 / m.

t1 = 0 ° C.

m = 1 kg.

U = 220 V.

T = 10 min = 600 s.

λ = 3.4 * 10 ^ 5 J / kg.

Efficiency = 60%.

L -?

According to the definition of efficiency = Ql * 100% / Qn, where Ql is the amount of heat required to melt ice, Qn is the amount of heat released in the heater.

Qн = U2 * T / R, where U is voltage, T is heater operation time, R is heater resistance.

R = ρ * L / S.

Qн = U2 * T * S / ρ * L.

Since the ice is at the melting temperature, Ql = λ * m, where λ is the specific heat of melting of ice, m is the mass of ice that has melted.

Efficiency = λ * m * 100% * ρ * L / U2 * T * S.

L = efficiency * U ^ 2 * T * S / λ * m * 100% * ρ.

L = 60% * (220 V) ^ 2 * 600 s * 0.5 mm2 / 3.4 * 10 ^ 5 J / kg * 1 kg * 100% * 1.1 Ohm * mm2 / m = 23.3 m …

Answer: the wire has a length L = 23.3 m.