Question

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).

Hint:

Only the radial velocity component carries mass through a cylindrical control surface at radius \(r\). Apply continuity, \(\rho(2\pi r b)v_r = \) constant, between inlet and outlet; the width \(b\) and density are constant so they cancel.

Answer 1

Correct Answer: 40

Step 1: Write the mass flow through a cylindrical surface using the diameter.
Instead of first converting to radius, keep the circumference in terms of diameter: circumference $= \pi D$, so the flow area at any diffuser station is $A = \pi D b$, with $b$ the constant width.

Step 2: Equate the radial mass flow at inlet and outlet.
$\dot{m} = \rho (\pi D_1 b) v_{r1} = \rho (\pi D_2 b) v_{r2}$. The factors $\rho$, $\pi$, and $b$ are identical on both sides (density and width both constant), so they cancel directly:
\[ D_1 v_{r1} = D_2 v_{r2} \]

Step 3: Substitute directly with the given diameters (no need to halve them into radii).
\[ (0.2)(60) = (0.3)(u) \]
\[ 12 = 0.3\,u \]

Final Answer:
\[ \boxed{u = 40 \text{ m/s}} \]