Question

Two cylindrical shafts A and B at the same initial temperature are simultaneously placed in a furnace. The surfaces of the shafts remain at the furnace gas temperature at all times after they are introduced into the furnace. The temperature variation in the axial direction of the shafts can be assumed to be negligible. The data related to shafts A and B is given in the following Table.

Quantity	Shaft A	Shaft B
Diameter (m)	0.4	0.1
Thermal conductivity (W/m-K)	40	20
Volumetric heat capacity (J/m3-K)	2 × 106	2 × 107

 

The temperature at the centreline of the shaft A reaches 400°C after two hours. The time required (in hours) for the centreline of the shaft B to attain the temperature of 400°C is ________ 

Answer 1

Concept:

Whenever the heat transfer coefficient (h) is infinite the heat transfer will occur even if there is no temperature difference between the two bodies.

If heat transfer coefficient (h) is infinity then the Biot number (Bi) will also be infinity since

Bi ∝ h

From Heisler’s chart, we know that

Temperature difference θ^* = f(s^*, Fo, Bi)

s^* = 0 at centerline.   .......(In Heisler's Chart)

Hence the only variable left is fourier no (Fo)

Equating fourier no for both cases

Calculation:

 ρ_A = 2 × 10⁶ J/m³-K, ρ_B = 2 × 10⁷ J/m³-K,  k_A = 40 W/mK, k_B  = 20 W/mK, L_A   = 0.4m, L_B = 0.1m, τA = 2 hrs

\(\frac{{{\alpha _A}\;{\tau _A}}}{{L_A^2}} = \frac{{{\alpha _B}\;{\tau _B}}}{{L_B^2}}\)

\(\frac{{{k_A}\;{\tau _A}}}{{{\rho _A}{C_p}{L_A^2}}} = \frac{{{k_B}\;{\tau _B}}}{{{\rho _B}{C_p}{L_B^2}}}\)

\(\left( {\frac{{40}}{{2 × {{10}^6} × {{0.4}^2}}}} \right) × 2 = \;\left( {\frac{{20}}{{2× {{10}^7} × {{0.1}^2}}}} \right) × {\tau _B}\)

\({\tau _B} = 2.5hrs\)

Points to remember:

• \(\frac{{\rho VC}}{{{h_A}}} = Time\;constant,\;unit \to second\)
• ρC = Volumetric heat capacity, unit → J/m³-K