Project ForgeHeat Transfer
Convection & Radiation
GATE 2015
Hard
For flow through a pipe of radius R, the velocity and temperature distribution are as follows: u(r,x)=C1, and T(r,x)=C2[1−(r/R)³], where C1 and C2 are constants. The bulk mean temperature is given by Tm=(2/(Um·R²))∫₀^R u(r,x)T(r,x)r dr, with Um being the mean velocity of flow. The value of Tm is
- A0.5C2/Um
- B0.5C2
- 0.6C2
- D0.6C2/Um
Explanation
Since u=C1 is uniform, Um=C1 too, so C1 cancels with Um in the formula: Tm=(2/R²)∫₀^R C2[1−(r/R)³]r dr = (2C2/R²)[R²/2 − R⁵/(5R³)] = (2C2/R²)[R²/2−R²/5] = 2C2(1/2−1/5) = 2C2(3/10) = 0.6C2.
Source: GATE ME 2015 (Session 1) (2015)
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