A centrifugal compressor has a constant-width radial diffuser. The diameters at the diffuser inlet and outlet are \(0.2\) m and \(0.3\) m, respectively. The flow at the diffuser inlet and outlet is assumed to be steady and uniform. The average velocity at the diffuser inlet and outlet are \((60\,\hat{e}_r + 75\,\hat{e}_{\theta})\) m/s and \((u\,\hat{e}_r + 50\,\hat{e}_{\theta})\) m/s, respectively. If the flow through the diffuser is treated as steady and incompressible, the value of \(u\) is _______ (rounded off to the nearest integer).
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Correct Answer: 40
Solution and Explanation
In a radial diffuser the flow crosses circular (cylindrical) surfaces centered on the shaft axis. The area of such a surface at radius \(r\) is \(A = 2\pi r b\), where \(b\) is the (constant) width of the diffuser. Only the radial velocity component \(v_r\) is normal to this surface, so only \(v_r\) carries mass flow through it; the tangential component \(v_\theta\) does not.
Step 2: Write continuity between inlet (station 1) and outlet (station 2).
For steady, incompressible flow (\(\rho\) constant) with constant width \(b\),
\[ \rho (2\pi r_1 b) v_{r1} = \rho (2\pi r_2 b) v_{r2} \]
The \(2\pi \rho b\) factor is the same on both sides and cancels, leaving
\[ r_1 v_{r1} = r_2 v_{r2} \]
Step 3: Substitute the given values.
\(r_1 = D_1/2 = 0.2/2 = 0.1\) m, \(v_{r1} = 60\) m/s (the \(\hat{e}_r\) component at inlet).
\(r_2 = D_2/2 = 0.3/2 = 0.15\) m, \(v_{r2} = u\) (the \(\hat{e}_r\) component at outlet, unknown).
\[ (0.1)(60) = (0.15)(u) \]
\[ 6 = 0.15\,u \]
Final Answer:
\[ \boxed{u = 40 \text{ m/s}} \]
(The tangential components 75 m/s and 50 m/s are not needed for this mass-continuity check; they would matter if the question asked about angular momentum or diffuser efficiency instead.)
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