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Chapter 7: Problem 12

The pressure rise, \(\Delta p,\) across a pump can be expressed as \\[ \Delta p=f(D, \rho, \omega, Q) \\] where \(D\) is the impeller diameter, \(\rho\) the fluid density, \(\omega\) the rotational speed, and \(Q\) the flowrate. Determine a suitable set of dimensionless parameters.

Short Answer

Expert verified
The dimensionless parameters for the given exercise are Pi1 = (Δp) / (ρ * (ω)² * (D)²) and Pi2 = Q / (ω * (D)³).

Step by step solution

01

Define the Dimensions of the Variables

Begin by delineating the physical dimensions of each variable in the equation. Let [D] denote the dimensions of D, [Q] denote the dimensions of Q etc. Here, D (diameter) has the dimension of length (L), ρ (density) has the dimension of mass per unit volume (ML⁻³), ω (angular velocity) has the dimension of T⁻¹ (time inversed), and Q (flowrate) has the dimension of volume per unit time (L³T⁻¹). Lastly, Δp (pressure difference), has the dimension of force per unit area (ML⁻¹T⁻²) or equivalently,
02

Apply the Buckingham Pi theorem

Now apply the Buckingham Pi theorem. This theorem states that any physical law can be expressed as a relationship among dimensionless quantities. As there are five variables (i.e., Δp, D, ρ, ω, Q) and three fundamental dimensions (M, L, T), the theorem suggests there should be 5 - 3 = 2 Pi terms.
03

Define the Pi terms

Let's define the Pi terms. The choice of which variables to build the Pi terms from is arbitrary, so pick the two which appear to be the simplest, which are D (Diameter) and ρ (density). Then look for two more variables such that (with D and ρ) they form a dimensionless group. Those groups can be expressed as follows: \[Pi1 = (Delta p) / (rho * (omega)^2 * (D)^2)\] \[Pi2 = Q / (omega * (D)^3)\]

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Most popular questions from this chapter

By inspection, arrange the following dimensional parameters into dimensionless parameters: (a) kinematic viscosity, \(v,\) length, \(\ell,\) and time, \(t ;\) and (b) volume flow rate, \(Q,\) pump diameter, \(D,\) and pump impeller rotation speed, \(N\)

Air bubbles discharge from the end of a submerged tube as shown in Fig. P7.57. The bubble diameter, \(D\), is assumed to be a function of the air flowrate, \(Q\), the tube diameter, \(d\), the acceleration of gravity, \(g\), the density of the liquid, \(\rho\), and the surface tension of the liquid, \(\sigma\). (a) Determine a suitable set of dimensionless variables for this problem. (b) Model tests are to be run on the Earth for a prototype that is to be operated on a planet where the acceleration of gravity is 10 times greater than that on Earth. The model and prototype are to use the same fluid, and the prototype tube diameter is 0.25 in. Determine the tube diameter for the model and the required model flowrate if the prototype flowrate is to be \(0.001 \mathrm{ft}^{3} / \mathrm{s}\).

A model hydrofoil is to be tested. Is it practical to satisfy both the Reynolds number and the Froude number for the hydrofoil when it is operating near the water surface? Support: your decision.

At a sudden contraction in a pipe the diameter changes from \(D_{1}\) to \(D_{2} .\) The pressure drop, \(\Delta p,\) which develops across the contraction is a function of \(D_{1}\) and \(D_{2},\) as well as the velocity, \(V\), in the larger pipe, and the fluid density, \(\rho,\) and viscosity, \(\mu .\) Use \(D_{1}, V,\) and \(\mu\) as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable?

As winds blow past buildings, complex flow patterns can develop due to various factors such as flow separation and interactions between adjacent buildings. (See Video \(\vee 7.13\).) Assume that the local gage pressure, \(p\), at a particular location on a building is a function of the air density, \(\rho,\) the wind speed, \(V\), some characteristic length, \(\ell,\) and all other pertinent lengths, \(\ell_{i},\) needed to characterize the geometry of the building or building complex. (a) Determine a suitable set of dimensionless parameters that can be used to study the pressure distribution. (b) An eight-story building that is \(100 \mathrm{ft}\) tall is to be modeled in a wind tunnel. If a length scale of 1: 300 is to be used, how tall should the model building be? (c) How will a measured pressure in the model be related to the corresponding prototype pressure? Assume the same air density in model and prototype. Based on the assumed variables, does the model wind speed have to be equal to the prototype wind speed? Explain.
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