Question

A cylinder of radius r is surrounded by a cylindrical shell of inner radius r and outer radius 2r. The thermal conductivity of the material of the inner cylinder is k1 and that of the outer cylinder is k2. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

Answer 1

Answer:

Step-by-step explanation:

Let R₁ = Initial Radius = R  

Let R₂ = Final Radius = 2R  

The two conductors are in parallel. Therefore, equivalent thermal resistance R is :-

1/R = 1/R₁ + 1/R₂ ---------- (1)

R= KA/l by Q = KA (T₁ - T₂)/ l

R₁= K₁A₁/l₁

R₂= K₂A₂/l₂

Substituting above in (1) we get,

KA/l = K₁A₁/l₁ + K₂A₂/l₂ ---------- (2)

Now, l₁ = l₂ = l

A₁ = ΠR²

A₂ = Π (2R)² - ΠR₂² = 3ΠR₂

A = Π (2R)² = 4ΠR²

Substituting in (2) we get,

K4ΠR²/ l = K₁ΠR²/l + K₂3ΠR₂/l

∴ K = (K₁ + 3K₂) /4