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Chapter 6: Problem 54

The flow in the impeller of a centrifugal pump is modeled by the superposition of a source and a free vortex. The impeller has an outer diameter of \(0.5 \mathrm{m}\) and an inner diameter of \(0.3 \mathrm{m}\). At the outlet from the impeller, the flowing water has the following velocity componeats, relative to the impeller: radial component \(2 \mathrm{m} / \mathrm{s}\) and tangential comporent \(7 \mathrm{m} / \mathrm{s}\). (a) Find the strength of the source and the vortex required to model this flow. (b) Assume that the impeller blades are shaped like the streamlines and plot an impeller blade shape. (c) Find the radial and tangential components of velocity at the inlet to the impeller.

Short Answer

Expert verified
The strength of the source is \(π\) m²/s, and that of the vortex is \(3.5π\) m²/s. The components of velocity at the inlet are \(V_{r,i} = 3.33\) m/s and \(V_{θ,i} = 23.33\) m/s.

Step by step solution

01

Calculate Strength of Source and Vortex

The fluid flow relative to the impeller is given by \(V_r = 2\) m/s and \(V_θ = 7\) m/s at the outlet with a radius \(r_o = 0.5/2 = 0.25\) m. The strength, S of the source is given by \(S = 2πrV_r\), and strength, Γ of the vortex is given by \(Γ = 2πrV_θ\). Substituting the given values into these formulas gives \(S = 2π × 0.25 × 2 = π\) m²/s and \(Γ = 2π × 0.25 × 7 = 3.5π\) m²/s.
02

Plot Impeller Blade Shape

Since the impeller blades are shaped like the streamlines, the flow path from inlet to outlet forms the blade shape. The blade shape would follow a spiral pattern since the flow is a combination of the source (radial flow) and free vortex (rotational flow). This won't be accurately depicted in this textual explanation, as it requires software and graphical interpretation.
03

Calculate Components of Velocity at Inlet

At the inlet with a radius \(r_i = 0.3/2 = 0.15\) m, the radial and tangential velocity components can be calculated using the strengths of source and vortex respectively. For the source, the radial component of velocity \(V_{r,i} = S / 2πr_i = π / (2π × 0.15) = 1/0.3 = 3.33\) m/s. For the vortex, the tangential component of velocity \(V_{θ,i} = Γ / 2πr_i = 3.5π / (2π × 0.15) = 7/0.3 = 23.33\) m/s.

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