A long slender bar has one of its end in a furnace and has reache

SOLUTION

Concept:

The slender bar could be considered as a long fin.

For long fin the temperature distribution follows

\(\frac{{T - \;{T_\infty }}}{{{T_0} - \;{T_\infty }\;}} = {e^{ - mx}}\)

Where, m = fin parameter = \(\sqrt {\frac{{hP}}{{kA}}} \)

T₀ = Furnace temperature, T_∞ = surrounding temperature

For same temperature at two locations we get,

mx = constant.

Solve for h by using formula for m

Calculation:

\(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = {e^{ - mx}}\)

mx = constant  (for same Temperature and x₁ and x₂)

m₁x₁ = m₂x₂

\({\left( {\sqrt {\frac{{hP}}{{kA}}} \;} \right)_1}{x_1} = {\left( {\sqrt {\frac{{hP}}{{kA}}} \;} \right)_2}\;{x_2}\)

Squaring on both the sides

∴ h₁ × x₁² = h₂ × x₂² (since all other parameters are same)

∴ h₁ (0.25)² = h₂ (0.15)²

∴ h₂ = 2.77 h₁

∴ n = 2.777.

∴ nⁿ = 17.080