This question was previously asked in

ISRO Refrigeration and Air Conditioning 2014 Official

Option 2 : 1.4

__ Concept__:

**Transient Conduction in Semi-Infinite Solid**:

\(\frac{T(x,t)\;-\;T_s}{T_i\;-\;T_s}=erf\left(\frac{x}{2\sqrt{α t}}\right)\)

where T_{i} = initial temperature, T(x,t) = temperature at location and time and T_{s} = surface temperature, α = thermal diffusivity, t = time and x = depth from surface.

__ Calculation__:

__ Given__:

T_{i} = 308 K, T_{s} = 328 K, T(x,t) = 318 K, α = 1.77 × 10-3 m2/hr and x = 5 cm = 0.05 m.

\(\frac{T(x,t)\;-\;T_s}{T_i\;-\;T_s}=\frac{318\;-\;328}{308\;-\;328}=0.5\)

From the graph we can see that the error function (erf) is linear up to 0.7, therefore

\(erf\left(\frac{x}{2\sqrt{α t}}\right)\approx\left(\frac{x}{2\sqrt{α t}}\right)\)

\(\frac{T(x,t)\;-\;T_s}{T_i\;-\;T_s}=\left(\frac{x}{2\sqrt{α t}}\right)=0.5\)

\(\left(\frac{x}{2\sqrt{α t}}\right)=0.5\)

\(\left(\frac{0.05}{2\sqrt{1.77\times 10^{-3}\times t}}\right)=0.5\)

\((0.05)^2=1.77 \times10^{-3}\times t\)

∴ t = 1.412 hr

ISRO Scientist ME 2020 Paper

1666

80 Questions
240 Marks
90 Mins