GATE 2022 ME (ME2) – Question 63
Consider steady state, one-dimensional heat conduction in an infinite slab of thickness 2L (L = 1 m) as shown in the figure. The conductivity (k) of the material varies with temperature as $k=CT$, where T is the temperature in K, and C is a constant equal to 2 W m$^{-1}$ K$^{-2}$. There is a uniform heat generation of 1280 kW/m$^3$ in the slab. If both faces of the slab are maintained at 600 K, then the temperature at x = 0 is ______ K (in integer).
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Correct answer: 1000
Explanation
**Step 1: heat flow by symmetry.** The slab is symmetric about $x=0$, so no heat crosses the centre plane ($q_x=0$ at $x=0$). With uniform generation $\dot q$, the heat flux at position $x$ is
$$q_x=\dot q\,x\quad(\text{outward}),\qquad q_x=-k\frac{dT}{dx}.$$
**Step 2: put in $k=CT$.**
$$-CT\frac{dT}{dx}=\dot q\,x\;\Rightarrow\;-C\,T\,dT=\dot q\,x\,dx .$$
**Step 3: integrate from the centre ($T_0$ at $x=0$) to the face ($T_s=600$ K at $x=L$):**
$$-\frac C2\left(T_s^2-T_0^2\right)=\frac{\dot q L^2}{2}\;\Rightarrow\;T_0^2-T_s^2=\frac{\dot qL^2}{C}.$$
**Step 4: numbers.**
$$\frac{\dot qL^2}{C}=\frac{1280\times10^3\times1^2}{2}=6.4\times10^5$$
$$T_0=\sqrt{600^2+640\,000}=\sqrt{1\,000\,000}=\mathbf{1000\ K}.$$