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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 = (螖p) / (蟻 * (蠅)虏 * (D)虏) and Pi2 = Q / (蠅 * (D)鲁).

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), 蟻 (density) has the dimension of mass per unit volume (ML鈦宦�), 蠅 (angular velocity) has the dimension of T鈦宦� (time inversed), and Q (flowrate) has the dimension of volume per unit time (L鲁T鈦宦�). Lastly, 螖p (pressure difference), has the dimension of force per unit area (ML鈦宦筎鈦宦�) 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., 螖p, D, 蟻, 蠅, 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 蟻 (density). Then look for two more variables such that (with D and 蟻) 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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Key Concepts

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

Buckingham Pi Theorem
The Buckingham Pi theorem is a key concept in dimensional analysis. It allows us to reduce complex physical phenomena into simpler, dimensionless parameters. Imagine having an equation with multiple units and variables. The theorem helps by identifying groups of terms that are dimensionally consistent. These are called Pi terms.

To use the theorem, we count the number of variables involved and subtract the number of fundamental dimensions (Mass, Length, Time, etc.) from them. This gives us the number of dimensionless groups or Pi terms to expect. In our problem, there are five variables: pressure rise \(\Delta p\), impeller diameter \(D\), fluid density \(\rho\), rotational speed \(\omega\), and flow rate \(Q\). With the fundamental dimensions of Mass \(M\), Length \(L\), and Time \(T\), the theorem predicts 2 Pi terms. This simplification helps in analysis, making complex equations easier to handle.
Dimensionless Parameters
Dimensionless parameters are powerful tools to understand and compare different physical systems. They are variables that, when combined, result in a number without units.

These parameters help highlight similarities between systems that might not be obvious when using traditional units. For example, in the given exercise, our aim is to express pressure rise across a pump through dimensionless groups.

The Pi terms derived are:
  • \(Pi_1 = \frac{\Delta p}{\rho \cdot \omega^2 \cdot D^2}\)
  • \(Pi_2 = \frac{Q}{\omega \cdot D^3}\)
Here, \(Pi_1\) and \(Pi_2\) are dimensionless, meaning their results are unitless numbers. They simplify the study of pump systems across different scales and conditions.
Fluid Mechanics
Fluid mechanics is a branch of physics concerned with the behavior of liquids and gases. It plays a crucial role in understanding systems like pumps. In fluid mechanics, we focus on properties like pressure, density, and flow rate.

Pumps are devices used to move fluids, and understanding their operation involves analyzing how these fluid properties interact. The dimensional analysis we conducted helps us see how factors like impeller diameter \(D\) and rotational speed \(\omega\) influence fluid behavior. As we study these interactions, we see how important it is to reduce complexity using dimensionless parameters.
Pressure Rise in Pumps
The pressure rise \(\Delta p\) in pumps is a key metric for evaluating pump performance. It represents the increase in pressure from the pump inlet to the pump outlet.

Several factors influence this value:
  • Diameter of the impeller \(D\)
  • Fluid density \(\rho\)
  • Rotational speed \(\omega\)
  • Flow rate \(Q\)
Understanding these factors helps in designing efficient pump systems. The dimensionless parameters \(Pi_1\) and \(Pi_2\) provide a way to study the effects of these variables together, helping engineers create pumps that perform well under various conditions.

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Most popular questions from this chapter

(See The Wide World of Fluids article "Modeling Parachutes in a Water Tunnel," Section \(7.8 .1 .\) ) Flow characteristics for a \(30-f t\) diameter prototype parachute are to be determined by tests of a 1-fit-diameter model parachute in a water tunnel. Some data collected with the model parachute indicate a drag of 17 lb when the water velocity is \(4 \mathrm{f}\) Us. Lse the model data to predict the drag on the prototype parachute falling through air at \(10 \mathrm{ft} / \mathrm{s}\). Assume the drag to be a function of the velocity, \(V\), the fluid density, \(\rho\), and the parachute diameter, \(D\).

The dimensional parameters used to describe the operation of a ship or airplane propeller (sometimes called a screw propeller) are rotational speed, \(\omega,\) diameter, \(D,\) fluid density, \(\rho\) speed of the propeller relative to the fluid, \(V\), and thrust developed, \(T .\) The common dimensionless groups are called the thrust coefficient and the advance ratio. Propose appropriate definitions for these groups.

An equation used to evaluate vacuum filtration is $$Q=\frac{\Delta p A^{2}}{\alpha\left(V R w+A R_{f}\right)}$$ Where \(Q \doteq L^{3} / T\) is the filtrate volume flow rate, \(\Delta p \doteq F / L^{2}\) the vacuum pressure differential, \(A \doteq L^{2}\) the filter area, \(\alpha\) the filtrate "viscosity," \(V \doteq L^{3}\) the filtrate volume, \(R \doteq L / F\) the sludge specific resistance, \(w \doteq F / L^{3}\) the weight of dry sludge per unit volume of filtrate, and \(R_{f}\) the specific resistance of the filter medium. What are the dimensions of \(R_{f}\) and and \(\alpha ?\)

For a certain model study involving a 1: 5 scale model it is known that Froude number similarity must be maintained. The possibility of cavitation is also to be investigated, and it is assumed that the cavitation number must be the same for model and prototype. The prototype fluid is water at \(30^{\circ} \mathrm{C}\), and the model fluid is water at \(70^{\circ} \mathrm{C}\). If the prototype operates at an ambient pressure of \(101 \mathrm{kPa}(\mathrm{abs}),\) what is the required ambient pressure for the model system?

The dimensionless parameters for a ball released and falling from rest in a fluid are $$C_{D}, \quad \frac{g t^{2}}{D}, \quad \frac{\rho}{\rho_{b}}, \quad \text { and } \quad \frac{V_{t}}{D}$$ where \(\left.C_{\mathbf{D}} \text { is a drag coefficient (assumed to be constant at } 0,4\right)\) \(g\) is the acceleration of gravity, \(D\) is the ball diameter, \(t\) is the time after it is released, \(\rho\) is the density of the fluid in which it is dropped, and \(\rho_{b}\) is the density of the ball. Ball 1 , an aluminum ball \(\left(\rho_{b_{1}}=2710 \mathrm{kg} / \mathrm{m}^{3}\right)\) having a diameter \(D_{1}=1.0 \mathrm{cm},\) is dropped in water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\right) .\) The ball velocity \(V_{\mathrm{t}}\) is \(0.733 \mathrm{m} / \mathrm{s}\) at \(t_{1}=\) 0.10 s. Find the corresponding velocity \(V_{2}\) and time \(t_{2}\) for ball 2 having \(D_{2}=2.0 \mathrm{cm}\) and \(\rho_{b_{2}}=\rho_{b_{1}} .\) Next, use the computed value of \(V_{2}\) and the equation of motion,$$\rho_{b} V_{g}-\frac{\mathrm{C}_{\mathrm{D}}}{2} \rho A V^{2}=\rho_{b} V \frac{d V}{d t}$$ where \(Y\) is the volume of the ball and \(A\) is its cross-sectional area. to verify the value of \(t_{2}\). Should the two values of \(t_{2}\) agree?
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