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Chapter 6: Problem 60

Kerosene at \(20^{\circ} \mathrm{C}\) flows through a section of 170 -mm- diameter pipeline with a velocity of \(2.5 \mathrm{~m} / \mathrm{s}\). The pressure loss in the section of pipeline is to be studied using a scale model that has a diameter of \(30 \mathrm{~mm}\), and water at \(20^{\circ} \mathrm{C}\) is to be used in the model study. What water velocity should be used in the model? If a pressure loss of \(15 \mathrm{kPa}\) is measured in the model, what is the corresponding pressure drop in the actual pipeline section?

Short Answer

Expert verified
Model velocity is about 0.441 m/s; pressure drop in pipeline is approximately 12.8 MPa.

Step by step solution

01

Identify Variables for the Prototype and the Model

For the prototype (actual pipeline), note that the diameter, \(D_p = 170\, \text{mm} = 0.17\, \text{m}\) and the velocity, \(V_p = 2.5\, \text{m/s}\). The model has a diameter \(D_m = 30\, \text{mm} = 0.03\, \text{m}\). The goal is to use similitude to find the model velocity, \(V_m\).
02

Apply Dynamic Similarity

Dynamic similarity implies that the Reynolds number for the prototype must equal that for the model. The Reynolds number is given by \(Re = \frac{\rho V D}{\mu}\). For dynamic similarity: \[Re_p = Re_m \] where \(\rho\) is density and \(\mu\) is the dynamic viscosity. Substituting for kerosene and water properties, we assume they are similar for a rough estimation, focusing primarily on diameter and velocity. Thus,\[ \frac{V_p D_p}{u_{kerosene}} = \frac{V_m D_m}{u_{water}} \]Where \(u\) represents the kinematic viscosity, given similar conditions.
03

Solve for the Model Velocity

Rearrange and calculate the model velocity: \[V_m = V_p \left( \frac{D_p}{D_m} \right) \left( \frac{u_{water}}{u_{kerosene}} \right)\]Assuming similar viscosity at similar temperatures and approximating,\(\frac{u_{water}}{u_{kerosene}} \approx 0.9\), we calculate:\[V_m \approx 2.5 \times \left( \frac{0.17}{0.03} \right) \times 0.9 \approx 0.441 \text{ m/s}\]
04

Calculate Corresponding Pressure Drop

To find the corresponding pressure drop in the pipeline, apply the proportionality in dynamic similarity:\[\left( \frac{\Delta P_p}{\Delta P_m} \right) = \left( \frac{D_p}{D_m} \right)^5\]Given \(\Delta P_m = 15 \text{ kPa}\), solve for \(\Delta P_p\):\[\Delta P_p = 15 \times \left( \frac{0.17}{0.03} \right)^5 \approx 15 \times 852.5 = 12787.5 \text{ kPa}\]It means the actual pressure drop in the prototype is approximately \(12.8 \text{ MPa}\).
05

Verify Units and Assumptions

Ensure that calculations maintain consistent unit usage and reasonable assumptions, including that viscosity conditions are accurately approximated. Review assumptions regarding fluid dynamic relations and approximated viscosities, ensuring that steps reflect expectations of real-world behavior between kerosene and water models.

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Key Concepts

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

Dynamic Similarity
Dynamic similarity is a fundamental principle in fluid dynamics used to ensure that experiments on scale models accurately represent real-world scenarios. It's about maintaining similar flow conditions between a prototype (the real system) and its model. This is achieved by ensuring that dimensionless numbers, like the Reynolds number, are the same in both the model and the prototype.

  • This concept allows engineers to use manageable, smaller models to predict the behavior of larger systems.
  • It relies on mathematical equations to ensure that the forces acting on the fluids in both the model and the prototype operate in a similar way.
Maintaining dynamic similarity helps to provide reliable predictions without needing to test full-scale systems directly. This is particularly useful in reducing costs and time associated with engineering experiments.
Reynolds Number
The Reynolds number \(Re\) is a dimensionless number pivotal in fluid dynamics. It helps predict flow patterns in different fluid flow situations. The Reynolds number is calculated using the formula \(Re = \frac{\rho VD}{\mu}\), where \(\rho\) is the fluid density, \(V\) is the fluid velocity, \(D\) is the characteristic length (like diameter), and \(\mu\) is the dynamic viscosity.

  • A high Reynolds number indicates turbulent flow, where fluid particles move chaotically.
  • A low Reynolds number signals laminar flow, characterized by smooth, orderly fluid motion.
  • For dynamic similarity, the Reynolds numbers of the prototype and the model must be identical.
In our exercise, ensuring the Reynolds numbers are the same for both the kerosene pipeline and the water model was key to accurately calculating the water velocity needed in the model study.
Pressure Drop
Pressure drop refers to the reduction in pressure as a fluid flows through a pipeline. It happens due to friction forces within the fluid and between the fluid and the pipe walls. In engineering and design, knowing the pressure drop is crucial for ensuring that pipelines and channels can handle fluid flow efficiently without losing too much energy.

  • The pressure drop is affected by factors like fluid velocity, pipe diameter, fluid viscosity, and pipe length and roughness.
  • In our exercise, the pressure drop in the scale model helped predict the corresponding drop in the actual prototype pipeline.
  • Dynamic similarity allows transferring the pressure drop findings from the scale model to the prototype, adjusted by a specific ratio.
Understanding pressure drop assists in designing effective fluid transport systems by avoiding unnecessary pressure losses.
Scale Model Study
A scale model study involves using a smaller, manageable version of a real-world system to investigate specific behaviors or properties of a fluid. This approach is essential in fluid dynamics, enabling detailed analysis and experiments on a scale model to make predictions about the actual prototype.

  • Scale models are designed to replicate the essential features of the prototype while being easier and more cost-effective to test.
  • The model must maintain dynamic similarity with the prototype to give accurate predictions.
  • In our exercise, a scale model of a pipeline using water instead of kerosene made it easier to study pressure drop behaviors.
Scale model studies are invaluable for engineering projects as they reduce the risks and costs associated with testing full-scale systems, ensuring efficient development and verification of fluid systems' designs.

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Most popular questions from this chapter

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