GATE 2023 CE (CE2) – Question 52
A 5 cm long metal rod AB was initially at a uniform temperature of $T_0$ °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 $\partial T/\partial t=D\partial^2T/\partial x^2$, where D is the thermal diffusivity of the metal, given as 1.0 cm$^2$/s. The temperature distribution in the rod is obtained as $T(x,t)=\sum_{n=1,3,5,\ldots}^{\infty}C_n\sin(n\pi x/5)e^{-\beta n^2t}$, where x is in cm measured from A to B with x = 0 at A, t is in s, $C_n$ 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 ______.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.385 to 0.405
Explanation
**Step 1: put one term of the series into the equation.** Take a single mode
$$T=\sin\!\left(\frac{n\pi x}{L}\right)e^{-\beta n^2t},\qquad L=5\text{ cm}.$$
**Step 2: derivatives.**
$$\frac{\partial T}{\partial t}=-\beta n^2\,T,\qquad\frac{\partial^2T}{\partial x^2}=-\left(\frac{n\pi}{L}\right)^2T .$$
**Step 3: insert into $\partial T/\partial t=D\,\partial^2T/\partial x^2$.**
$$-\beta n^2T=-D\left(\frac{n\pi}{L}\right)^2T\;\Rightarrow\;\beta=\frac{D\pi^2}{L^2}.$$
**Step 4: numbers.**
$$\beta=\frac{1.0\times\pi^2}{25}=0.3948\approx\mathbf{0.395\ s^{-1}}.$$