/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 An orifice plate is a flat plate... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 20

An orifice plate is a flat plate with a central opening that is sometimes used to measure the volume flow rate in a pipe. The pressure drop across an orifice plate can be assumed to be a function of the diameter of the opening in the plate, the diameter of the pipe, the velocity of flow in the pipe, and the density and viscosity of the fluid. Use dimensional analysis to determine the functional relationship between the pressure drop across the plate and the influencing variables. Express this functional relationship in both a dimensional and nondimensional form. Identify any named conventional dimensionless groups that appear in the nondimensional relationship.

Short Answer

Expert verified
The nondimensional form is \( \frac{\Delta P}{\rho V^2} = \phi\left(\frac{d}{D}, Re\right) \), incorporating the Reynolds number (Re).

Step by step solution

01

Understand the Problem

The problem involves finding a functional relationship using dimensional analysis between the pressure drop across an orifice plate and its influencing variables, namely: diameter of the opening (\(d\)), diameter of the pipe (\(D\)), velocity of flow (\(V\)), density of the fluid (\(\rho\)), and viscosity of the fluid (\(\mu\)).
02

List Relevant Variables and Dimensions

Determine the dimensions of each involved variable:- Pressure Drop (\(\Delta P\)): \([ML^{-1}T^{-2}]\)- Opening Diameter (\(d\)): \([L]\)- Pipe Diameter (\(D\)): \([L]\)- Velocity (\(V\)): \([LT^{-1}]\)- Density (\(\rho\)): \([ML^{-3}]\)- Viscosity (\(\mu\)): \([ML^{-1}T^{-1}]\)
03

Perform Dimensional Analysis

Using Pi theorem, let:\[ \Delta P = f(d, D, V, \rho, \mu) \]Assume\[ \Delta P = K d^a D^b V^c \rho^d \mu^e \]Substitute dimensions:\[ [ML^{-1}T^{-2}] = [L]^a[L]^b[LT^{-1}]^c[ML^{-3}]^d[ML^{-1}T^{-1}]^e \]Balance dimensions for \( M, L, \) and \( T \).
04

Solve for Unknowns

Balancing the equations:- Mass (\(M\)): 1 = d + e- Length (\(L\)): -1 = a + b + c - 3d - e- Time (\(T\)): -2 = -c - eSolving, we find:\(a = 1, b = 0, c = 2, d = 1, e = 1\)Hence,\[ \Delta P = f(dV^2\rho, \mu) \]
05

Express Nondimensional Form

Introduce the Reynolds number \(Re\) and use the functional relationship:\[ Re = \frac{\rho V D}{\mu} \]Substitute back to form dimensionless terms:\[ \frac{\Delta P}{\rho V^2} = \phi\left(\frac{d}{D}, Re\right) \]
06

Identify Named Dimensionless Groups

The Reynolds number \(Re\) is a conventional dimensionless group that summarizes inertial forces to viscous forces which appears in the nondimensional form of the equation.

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Key Concepts

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

Orifice Plate
An orifice plate is a simple and ingenious device used in fluid mechanics to measure flow rates in pipes. It's essentially a thin, flat plate with a hole in the center, and it's inserted into the pipe perpendicular to the flow of fluid. The size of this hole, or the opening diameter, is crucial as it directly affects the flow characteristics. When fluid flows through the pipe and reaches the orifice plate, it has to force itself through the smaller area of the hole. This change in area causes a difference in pressure, known as the pressure drop, which can be used to infer flow rates. The orifice plate's effectiveness and the pressure drop it causes depend on several factors such as the pipe's diameter, the fluid's velocity, and its physical properties like density and viscosity.
Pressure Drop
In fluid systems, a pressure drop occurs when there is a decrease in fluid pressure from one point to another. This is a common occurrence when the fluid passes through an obstruction, such as an orifice plate.
The pressure drop is influenced by multiple factors including:
  • The diameter of the orifice opening
  • The flow velocity of the fluid
  • Properties of the fluid such as density and viscosity
Understanding the pressure drop is essential because it helps in determining the flow rate within the system. The greater the pressure drop for a given flow rate, the more effective the orifice plate is in measuring flow. The relationship between these variables can be precisely analyzed using dimensional analysis.
Reynolds Number
The Reynolds number (Re) plays a crucial role in fluid dynamics, as it helps to predict flow patterns in different fluid flow situations. It's a dimensionless number that reflects the ratio of inertial forces to viscous forces within the fluid flow. The formula to calculate the Reynolds number is: \[ Re = \frac{\rho V D}{\mu} \]Where:
  • \( \rho \) represents the fluid density
  • \( V \) is the fluid velocity
  • \( D \) is the diameter of the pipe
  • \( \mu \) is the dynamic viscosity of the fluid
A higher Reynolds number indicates turbulent flow, while a lower number suggests laminar flow. Understanding and calculating the Reynolds number allows engineers to predict flow behaviors using dimensionless expressions, which is critical for designing systems with orifice plates and analyzing pressure drops.
Flow Rate Measurement
Flow rate measurement is essential for operations that involve fluid transport. It quantifies the volume of fluid passing through a point in a system per unit of time. In the case of an orifice plate, the pressure drop across the plate is related to the flow rate of the fluid.
Dimensional analysis can be used to express this relationship mathematically, helping to derive expressions that predict how changes in different variables affect the flow rate. By examining how the pressure drop correlates with factors such as pipe and orifice diameter, velocity of the fluid, and the fluid鈥檚 properties, engineers can accurately measure and control flow rates. This information is then employed in various applications鈥攆rom water management to oil refining鈥攈ighlighting the significance of precise flow rate measurement in ensuring the efficiency and safety of fluid handling systems.

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Most popular questions from this chapter

The shear stress, \(\tau_{0}\), exerted on a pipe of diameter, \(D,\) by a fluid of density, \(\rho,\) and viscosity, \(\mu\), moving with a velocity, \(V,\) is given by the following dimensionless relation: $$\frac{\tau_{0}}{\rho V^{2}}=f\left(\frac{\rho V D}{\mu}, \frac{\epsilon}{D}\right)$$ where \(\epsilon\) is the height of the roughness elements on the surface of the pipe. In a prototype pipe, the velocity of flow is \(2 \mathrm{~m} / \mathrm{s}\), the diameter is \(3 \mathrm{~m},\) and the height of the roughness elements is \(2 \mathrm{~mm}\). If a model of the pipe is to be constructed based on Reynolds number similarity (because the viscous force is important), a model scale of 1: 20 is used, and the same fluid is used in both the model and prototype, what velocity and roughness height should be used in the model? If the shear stress on the pipe surface in the model is \(2.25 \mathrm{kPa}\), what is the corresponding shear stress in the prototype?

A fluid of density \(\rho\) and dynamic viscosity \(\mu\) flows with a velocity \(V\) toward a rectangular plate of width \(W\), height \(H\), and thickness \(T\). The approaching fluid flow makes an angle \(\theta\) with the direction normal to the plate. Determine the functional relationship between dimensionless groups that would be appropriate for studying the relationship between the drag force on the plate and the given independent variables.

A two-dimensional flow field in the \(x y\) plane has velocity components \(u\) and \(v\) in the \(x\) - and \(y\) -directions, respectively, and the density, \(\rho\), of the fluid varies in both space \((x, y)\) and time, \(t\). The continuity equation for this two-dimensional compressible flow is given by $$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)=0$$ The velocity, length, time, and density scales in the flow field are \(V, L, T,\) and \(\rho_{0},\) respectively. It is common practice to represent the time scale, \(T,\) in terms of a frequency scale, \(\omega\), where \(\omega=2 \pi / T\); in such cases, the time scale becomes \(\omega^{-1}\) (a) Express Equation 6.28 in normalized form. Use the Strouhal number, St, defined as \(\mathrm{St}=\omega L / V\), in the final expression. (b) Give the asymptotic form of the normalized continuity equation as the Strouhal number becomes small and state a physical scenario in which the Strouhal number might become negligibly small.

Consider the energy equation for flow of a fluid in a closed conduit, which can be expressed in the form $$\frac{p_{1}}{\gamma}+\frac{V_{1}^{2}}{2 g}+z_{1}=\frac{p_{2}}{\gamma}+\frac{V_{2}^{2}}{2 g}+z_{2}+h_{\mathrm{f}}$$ where \(p_{1}, V_{1}\), and \(z_{1}\) are the pressure, average velocity, and centerline elevation at an upstream section of the conduit, \(p_{2}, V_{2},\) and \(z_{2}\) are the corresponding variables at a downstream section, \(h_{f}\) is the head loss between the two sections, \(\gamma\) is the specific weigh of the fluid, and \(g\) is the gravity constant. State the SI unit of energy and state the SI unit(s) of the terms (not the individual variables) in Equation 6.27. Explain why each of the terms in Equation 6.27 represents energy and express Equation 6.27 as a nondimensional equation.

A pump impeller of diameter \(D\) contains a mass \(m\) of fluid and rotates at an angular velocity of \(\omega\). Use dimensional analysis to obtain a functional expression for the centrifugal force \(F\) on the fluid in terms of \(D, m,\) and \(\omega\).
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