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Chapter 7: Problem 50

For a certain fluid flow problem it is known that both the Froude number and the Weber number are important dimensionless parameters. If the problem is to be studied by using a 1: 15 scale model, determine the required surface tension scale if the density scale is equal to \(1 .\) The model and prototype operate in the same gravitational field.

Short Answer

Expert verified
The required surface tension scale is 1:15. Therefore, the surface tension in the model should be 1/15 of the surface tension in the prototype.

Step by step solution

01

Identify Known Parameters

The given parameters are: the scale of the model which is 1:15, the Froude number, the Weber number, and the surface tension scale which we need to determine. The density scale is equal to 1 and both the model and prototype operate under the same gravitational field.
02

Apply the Froude Number

The Froude number (F) is defined as \(F = \frac{U}{\sqrt{gL}}\), where \(U\) is the flow velocity, \(g\) is the acceleration due to gravity, and \(L\) is a characteristic length. From the given information, the Froude number for the model (F_m) is the same as that for the prototype (F_p). Hence, \(F_m = F_p\) or \(U_m/\sqrt{g_mL_m} = U_p/\sqrt{g_pL_p}\). As they operate in the same gravitational field \(g_p = g_m = g\), we get the ratio \(U_m/U_p = \sqrt{L_m/L_p}\), therefore, the velocity scales as the square root of the linear dimension.
03

Apply the Weber Number

The Weber number (We) is defined as \(We = \frac{\rho U^2 L}{\sigma}\), where \(U\) is the velocity, \(L\) is a characteristic length, \(\rho\) is the density, and \(\sigma\) is the surface tension. From the given information, the Weber number for the model (We_m) is the same as that for the prototype (We_p). Hence, \(We_m = We_p\) or \(\frac{\rho_m U_m^2 L_m}{\sigma_m} = \frac{\rho_p U_p^2 L_p}{\sigma_p}\). Given that the density scale \(\rho_m/\rho_p = 1\), we can simplify this to \(U_m^2/\sigma_m = U_p^2/\sigma_p\). Substituting the scale from step 2 for \(U_m/U_p=\sqrt{L_m/L_p}\), we get the relation \(\sqrt{L_m/L_p}^2/\sigma_m = \sqrt{L_m/L_p}^2/\sigma_p\), which simplifies to \(\sigma_m = \sigma_p (L_m/L_p)\).
04

Calculate Scale for Surface Tension

By applying the ratio of the scale model 1:15, we substitute the corresponding values into the formula derived from step 3 to give \(\sigma_m = \sigma_p (1/15)\). This gives us the scale for the surface tension.

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Key Concepts

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

Froude number
The Froude number is a fundamental dimensionless parameter used primarily in the study of fluid dynamics. It expresses the ratio of inertial forces to gravitational forces in a flow system. Formally, it is given by the formula: \[ F = \frac{U}{\sqrt{gL}} \]where:- \( U \) is the flow velocity,- \( g \) is the acceleration due to gravity,- \( L \) is a characteristic length of the system.
The Froude number is particularly significant in scenarios where fluids flow with a free surface, such as in rivers and seas. It helps in assessing wave and hydraulic jump behaviors.
In the case of scale models, keeping the Froude number consistent between the model and the prototype ensures that the flow dynamics remain similar. This is crucial for accurately studying phenomena like wave propagation or ship hull resistance in model testing environments.
Weber number
The Weber number is another essential dimensionless parameter, especially in situations involving surface tension. It is defined by the relation:\[ We = \frac{\rho U^2 L}{\sigma} \]where:- \( \rho \) is the fluid density,- \( U \) is the velocity,- \( L \) denotes a characteristic length,- \( \sigma \) is the surface tension.
This parameter helps understand the balance between the inertial forces and the surface tension forces in a fluid flow. It becomes significant in analyzing the behavior of droplets, bubbles, and liquid jets.
In scale modeling, maintaining an equivalent Weber number between the model and the prototype ensures that the effects of surface tension on the flow behavior are consistently represented across different scales. This balance is especially critical when surface phenomena like capillary waves are to be studied.
Surface tension scale
Surface tension, denoted by \( \sigma \), is the measure of the cohesive forces at the surface of a fluid. It plays a key role in many physical phenomena, from the formation of droplets to the movement of small objects on a liquid surface.
In a scale model analysis, it's vital to adjust the surface tension to match the effects seen in the prototype. This adjustment maintains the geometrical similarity of the flow path and the physical realism of the model.By ensuring that the surface tension scales appropriately with the model dimensions, as highlighted in the exercise above, we ensure the model effectively replicates the surface interaction behaviors of the prototype. Adjusting the surface tension requires precise calculation; in this case, it scales with the linear dimension of the model to the prototype ratio.
Scale model analysis
Scale model analysis is a technique widely used in engineering to study complex systems. It involves creating a scaled-down version of the actual system (prototype) to examine behavioral and physical properties without the cost or complexity of full-scale testing.
A successful scale model analysis relies heavily on maintaining dimensionless parameters, such as the Froude number and Weber number, identical between the model and the prototype.
The critical step in scale modeling is identifying the key dimensionless numbers that govern the physical phenomena of interest. By keeping these numbers consistent, engineers can be confident that the model faithfully replicates the prototype's behavior. For fluid dynamics, it often involves modifying parameters such as velocity, length, and surface tension according to precise scaling laws, as demonstrated in the exercise solution. This approach enables the analysis and design of systems like dams, bridges, and airfoils accurately and efficiently.

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Most popular questions from this chapter

A mixing basin in a sewage filtration plant is stirred by mechanical agitation (paddlewheel) with a power input \(\dot{W}(\mathrm{ft} \cdot \mathrm{lb} / \mathrm{s})\) The degree of mixing of fluid particles is measured by a "velocity gradient" \(G\) given by $$G=\sqrt{\frac{\dot{W}}{\mu V}}$$ where \(\mu\) is the fluid viscosity in \(\mathrm{Ib} \cdot \mathrm{s} / \mathrm{ft}^{2}\) and \(\mathrm{V}\) is the basin volume in \(\mathrm{ft}^{3}\). Find the units of the velocity gradient.

The following dimensionless groups are often used to present data on centrifugal pumps: flow coefficient \(\varphi=\frac{Q}{\omega D^{3}},\) head coefficient \(\psi=\frac{g H}{\omega^{2} D^{2}},\) power coefficient \(\xi=\frac{\dot{W}}{\omega^{3} D^{5}},\) efficiency \(r_{i}=\) \frac{\rho g Q H}{\dot{W}}, \text { specific speed } N_{s}=\frac{\omega \sqrt{Q}}{(g h)^{3 / 4}}, \text { specific diameter } D_{s}=\frac{\omega \sqrt{Q}}{(g h)^{1 / 4}} Show that the last three groups can be formed from combinations of the first three groups.

A coach has been trying to evaluate the accuracy of a baseball pitcher. After two years of studying, he proposes a function that can be presented as the accuracy of any pitcher: \\[ \mathrm{Acc}=f(V, a, m, \rho, p, z) \\] where Acc is the dimensionless accuracy, \(V\) is the velocity of the ball, \(a\) is the age of the pitcher, \(m\) is the mass of the pitcher, \(\rho\) is the density of air where the game is played, \(p\) is the pressure where the game is played (which varies with elevation), and \(z\) is the elevation above sea level. Find the dimensionless groups for this function.

The basic equation that describes the motion of the fluid above a large oscillating flat plate is $$\frac{\partial u}{\partial t}=v \frac{\partial^{2} u}{\partial y^{2}}$$ where \(u\) is the fluid velocity component parallel to the plate, \(t\) is time, \(y\) is the spatial coordinate perpendicular to the plate, and \(v\) is the fluid kinematic viscosity. The plate oscillating velocity is given by \(U=U_{0} \sin \omega t .\) Find appropriate dimensionless parameters and the dimensionless differential equation.

A dam spillway is \(40 \mathrm{ft}\) long and has fluid velocity of \(10 \mathrm{ft} / \mathrm{s}\) Considering Weber number effects as minor, calculate the corresponding model fluid velocity for a model length of \(5 \mathrm{ft}\).
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