GATE Civil Engineering (CE) 2023 Shift 2 Solved Paper

A 5 cm long metal rod AB was initially at a uniform temperature of T0°C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation $\frac{\partial \mathit{T}}{\partial \mathit{t}}=\mathit{D}\frac{{\partial }^2\mathit{T}}{\partial {\mathit{x}}^2}$, where D is the thermal diffusivity of the metal, given as 1.0 cm2/s.
The temperature distribution in the rod is obtained as
$\mathit{T}\left(\mathit{x},\mathit{t}\right)={\mathit{\Sigma }}_{\mathit{n}=1,3,5...}^{\mathit{\infty }}{\mathit{C}}_{\mathit{n}}\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{n}\mathit{\pi }\mathit{x}}{5}{\mathit{e}}^{-\mathit{\beta }{\mathit{n}}^2\mathit{t}}$,
where x is in cm measured from A to B with 𝑥 = 0 at A, t is in s, 𝐶𝑛 are constants in °C, T is in °C, and β is in s−1 .
The value of β (in 𝑠−1 , rounded off to three decimal places) is ________.