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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 \(蟺\) m虏/s, and that of the vortex is \(3.5蟺\) m虏/s. The components of velocity at the inlet are \(V_{r,i} = 3.33\) m/s and \(V_{胃,i} = 23.33\) 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 \(V_胃 = 7\) 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 \(S = 2蟺rV_r\), and strength, 螕 of the vortex is given by \(螕 = 2蟺rV_胃\). Substituting the given values into these formulas gives \(S = 2蟺 脳 0.25 脳 2 = 蟺\) m虏/s and \(螕 = 2蟺 脳 0.25 脳 7 = 3.5蟺\) m虏/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 \(V_{r,i} = S / 2蟺r_i = 蟺 / (2蟺 脳 0.15) = 1/0.3 = 3.33\) m/s. For the vortex, the tangential component of velocity \(V_{胃,i} = 螕 / 2蟺r_i = 3.5蟺 / (2蟺 脳 0.15) = 7/0.3 = 23.33\) m/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Source Strength in Fluid Mechanics
In fluid mechanics, understanding the concept of source strength is crucial when analyzing centrifugal pump systems. Source strength determines how much fluid is being added to the system per unit time. It assists in defining radial flow from the center artifact.
In our problem scenario, the source strength is represented by the velocity component radially, denoted by \(V_r\). It is calculated using the formula \(S = 2\pi r V_r\), where \(r\) indicates the radial distance from the center. For the impeller described, this radial component is 2 m/s at the outlet, with the radius at 0.25 m.
  • Application in Systems: This principle applies in many mechanical systems beyond pumps, helping design processes in which fluid dispersion from a point is essential.
  • Crucial Parameters: Distance from the source and velocity at this distance heavily impact the radial outflow profile.

Understanding source strength is important for predicting how fluid moves through centrifugal pump components and optimizing their performance.
Vortex Flow Modeling
Vortex flow modeling is a core technique used to predict the movement of fluids in rotational domains, such as in impeller systems of centrifugal pumps. Vortex flow involves a circular pattern where fluid elements follow a path similar to whirlpools.
Our example introduced a free vortex where tangential velocity \(V_胃\) at the outlet is specified to be 7 m/s. This characteristic describes how strongly the fluid rotates around the impeller. The strength of this vortex, \(\Gamma\), is crucial for performing vortex flow analysis.
  • Mathematical Formula: This is expressed by \(\Gamma = 2\pi r V_胃\).
  • Applications: Vortices occur naturally in rotational devices and need to be accurately modeled to ensure stable and efficient operation.

Free vortex models, as used in this impeller scenario, provide insights into controlling rotational fluid flows, minimizing turbulence, and maximizing transmission efficiency.
Impeller Blade Design
Designing impeller blades involves careful consideration of streamlining fluid flow, aiding in translating velocity into efficient pressure heads. The shape of the blades mimics the flow path of fluid from inlet to outlet, adeptly managing radial and tangential flow components.

This exercise assumes impeller blades mimic streamlines formed from the combined flow of source and vortex.
  • Flow Characteristics: Streamlining involves crafting spiral patterns, reflecting how the fluid actually passes through the impeller system.
  • Efficiency Optimization: Correct blade shapes minimize energy loss, achieving ideal flow separation and minimum stalling.
Good impeller blade design mitigates pressure losses, optimizes flow efficiency, and ensures longevity of pump function. Proper blade shape contributes significantly not just to the performance but also to the reliability of centrifugal pumps.

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Most popular questions from this chapter

(a) Show that for Poiseuille flow in a tube of radius \(R\) the magnitude of the wall shearing stress, \(\tau_{r}\), can be obtained from the relationship $$\left|\left(\tau_{r_{z}}\right)_{\mathrm{will}}\right|=\frac{4 \mu Q}{\pi R^{3}}$$ for a Newtonian fluid of viscosity \(\mu .\) The volume rate of flow is \(Q\) (b) Determine the magnitude of the wall shearing stress for a fluid having a viscosity of \(0.004 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) flowing with an average velocity of \(130 \mathrm{mm} / \mathrm{s}\) in a \(2-\mathrm{mm}\) -diameter tube.

Two horizontal, infinite, parallel plates are spaced a distance \(b\) apart. A viscous liquid is contained between the plates. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity \(U\). Because of the no-slip boundary condition (see Video \(V 6.12\) ), the liquid motion is caused by the liquid being dragged along by the moving boundary. There is no pressure gradient in the direction of flow. Note that this is a so-called simple Couette flow discussed in Section 6.9 .2 (a) Start with the Navier-Stokes equations and determine the velocity distribution between the plates. (b) Determine an expression for the flowrate passing between the plates (for a unit width). Express your answer in terms of \(b\) and \(U\).

The \(x\) -velocity profile in a certain laminar boundary layer is approximated as follows $$u=U_{0} \sin \left(\frac{\pi}{2} \frac{y}{0.1 \sqrt{x}}\right)$$ Determine the \(y\) -velocity, \(v(x, y)\).

Blood flows at volume rate \(Q\) in a circular tube of radius \(R\). The blood cells concentrate and flow near the center of the tube. while the cell-free fluid (plasma) flows in the outer region. The center core of radius \(R_{c}\) has a viscosity \(\mu_{c}\) and the plasma has a viscosity \(\mu_{p}\). Assume laminar, fully developed flow for both the core and plasma flows and show that an "apparent" viscosity is defined by $$\mu_{\mathrm{app}}=\frac{\pi R^{4} \Delta p}{8 L Q}$$ is given by $$\mu_{\mathrm{app}}=\frac{\mu_{p}}{1-\left(R_{c} / R\right)^{4}\left(1-\mu_{p} / \mu_{c}\right)}$$

A Bingham plastic is a fluid in which the stress \(\tau\) is related to the rate of strain \(d u / d y\) by $$\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)$$ where \(\tau_{0}\) and \(\mu\) are constants. Consider the flow of a Eingham plastic between two fixed, horizontal, infinitely wide, flat plates. For fully developed flow with \(d u / d x=0\) and \(d p / d x\) constant. find \(u(y)\).
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