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GATE CE 2023 Set 2 — Question 52

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NAT+2 / -0Medium1D Diffusion EquationPartial Differential EquationsEngineering Mathematics

Engineering Mathematics → Partial Differential Equations → 1D Diffusion Equation

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Question

A 5 cm long metal rod AB was initially at a uniform temperature of T0T_0 °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 ∂T∂t=D∂2T∂x2\frac{\partial T}{\partial t} = D \frac{\partial^2 T}{\partial x^2}, where DD is the thermal diffusivity of the metal, given as 1.0 cm2^2/s.
The temperature distribution in the rod is obtained asT(x,t)=∑n=1,3,5…∞Cnsin⁡nπx5e−βn2t,T(x,t) = \sum_{n=1,3,5\dots}^{\infty} C_n \sin \frac{n\pi x}{5} e^{-\beta n^2 t},where xx is in cm measured from A to B with x=0x = 0 at A, tt is in s, CnC_n are constants in °C, TT is in °C, and β\beta is in s−1^{-1}.
The value of β\beta (in s−1^{-1}, rounded off to three decimal places) is _____________.
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