Question

A cylindrical wire has a radius R and a thermal conductivity Kw. The fluid outside the wire has a mean temperature To and heat transfer coefficient

ℎ (assumed to be constant). The wire gets hot when it carries current because of its electrical resistance. The rate of thermal energy generation
may be expressed as
Qₛ = i²/kₑrho
where i is the current density (amps/m²),kₑ is the wire electrical conductivity (1/o hm**m), and rho is the wire mass density.

What is the dimensionless temperature distribution in the wire? What is the dimensionless rate of energy loss from the wire?

Answer 1

Dimensionless temperature distribution:

θ(r) = (T(r) - Tₒ)/(T₀ - Tₒ)

Dimensionless rate of energy loss:

qₛ / (hA(T₀ - Tₒ)) = 1 / Bi

Bi = hR / kₒₖ

The dimensionless temperature distribution (θ(r)) is a normalized representation of temperature within the wire at a given radial position r. It is calculated as θ(r) = (T(r) - Tₒ)/(T₀ - Tₒ), where T(r) is the temperature at radial position r, T∞ is the ambient temperature, and T0 is the initial temperature.

The dimensionless rate of energy loss is expressed as qₛ / (hA(T₀ - Tₒ)) = 1 / Bi, where qs​ is the rate of thermal energy generation, h is the heat transfer coefficient, A is the cross-sectional area of the wire, and Bi is the Biot number. The Biot number (Bi) is defined as the ratio of convective heat transfer to conductive heat transfer, calculated as hR/kw, where R is the radius of the wire and kw is the thermal conductivity of the wire material. The Biot number provides insight into whether convection or conduction dominates the overall heat transfer process in the system.