GATE 2021 CH – Question 58
As shown in the figure, air flows in parallel to a freshly painted solid surface of width 10 m, along the z-direction. The equilibrium vapor concentration of the volatile component A in the paint, at the air-paint interface, is $C_{A,i}$. The concentration $C_A$ decreases linearly from this value to zero along the y-direction over a distance $\delta=0.1$ m in the air phase. Over this distance, the average velocity of the air stream is 0.033 m/s and its velocity profile (in m/s) is $v_z(y)=10y^2$, where y is in meter. Let $C_{A,m}$ represent the flow averaged concentration. The ratio of $C_{A,m}$ to $C_{A,i}$ is _____ (round off to 2 decimal places).
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Correct answer: 0.24 to 0.26
Explanation
The concentration profile is $C_A(y)=C_{A,i}(1-y/\delta)$. A **flow-weighted** mean is $C_{A,m}=\int_0^\delta v_z C_A\,dy/\int_0^\delta v_z\,dy$; the 10 m surface width cancels.
Using $v_z=10y^2$,
$$\frac{C_{A,m}}{C_{A,i}}=\frac{10\int_0^\delta(y^2-y^3/\delta)\,dy}{10\int_0^\delta y^2\,dy}=\frac{\delta^3(1/3-1/4)}{\delta^3/3}=\mathbf{0.25}.$$
Using the rounded mean velocity 0.033 m/s instead gives 0.2525, which has the same two-decimal answer. An unweighted concentration average of 0.5 would be incorrect.