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Chapter 3: Problem 42

Under severe conditions, the wind speed at a particular location is \(35 \mathrm{~m} / \mathrm{s}\) and the pressure is \(101 \mathrm{kPa}\). At a downwind location, the wind flows between buildings, thereby increasing its velocity. Assuming that the density of air is constant as long as the pressure does not vary by more than \(10 \%,\) determine the maximum betweenbuilding velocity for which the incompressible Bernoulli equation can be applied. Would you expect such velocities between buildings under hurricane conditions? Explain.

Short Answer

Expert verified
Maximum between-building velocity is approx 52.15 m/s. Such velocities can occur under hurricane conditions.

Step by step solution

01

Understand Bernoulli's Equation

Bernoulli's equation, for incompressible flow, is given by: \[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \] where \(P\) is the pressure, \(\rho\) is the density, and \(v\) is the velocity of the fluid. Given that the pressure change is within 10%, we can use this equation to find the maximum velocity between buildings.
02

Establish Known Variables

From the problem: \(v_1 = 35 \mathrm{~m/s}\), \(P_1 = 101 \mathrm{kPa}\). As the pressure change should be within 10%, the permissible range for \(P_2\) is between 90.9 kPa and 111.1 kPa.
03

Bernoulli鈥檚 Equation Rearrangement

Rearrange Bernoulli's equation to solve for \(v_2\): \[ v_2^2 = v_1^2 + \frac{2(P_1 - P_2)}{\rho} \] Assuming air density \(\rho = 1.225 \mathrm{~kg/m^3}\) (standard condition density), we'll solve for \(v_2\) when speed is maximum and pressure is minimum.
04

Calculate Maximum Velocity

Given \(P_2 = 90.9 \mathrm{kPa}\), substitute the values into the equation: \[ v_2 = \sqrt{v_1^2 + \frac{2(P_1 - P_2)}{\rho}} \] \[ v_2 = \sqrt{35^2 + \frac{2(101000 - 90900)}{1.225}} \] Calculate \(v_2\) to find the maximum velocity.
05

Compute Velocity

Carry out the computation: \[ v_2 = \sqrt{1225 + \frac{2 \cdot 10100}{1.225}} \] Simplifying gives \(v_2 \approx 52.15 \mathrm{~m/s}\).
06

Assess Realistic Conditions

In hurricane situations, wind speeds can exceed even this calculated maximum. Thus, while theoretically possible, high velocities between buildings in such conditions could exceed the assumption limits of Bernoulli鈥檚 equation.

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Key Concepts

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

Fluid Dynamics
Fluid dynamics is a branch of physics that studies the movement of fluids鈥攚hether liquids or gases鈥攁nd the forces acting on them. This field is essential for understanding how air and water flow in various scenarios, from weather patterns to aircraft aerodynamics. In the case of Bernoulli's equation, it applies to fluid dynamics by showing the relationship between pressure, velocity, and density in a flowing fluid.
Bernoulli's equation tells us that in a streamline flow, the sum of pressure energy, kinetic energy, and potential energy remains constant. This is particularly useful in predicting how fluid behaves in different conditions.
  • Streamline flow: A flow in which all fluid particles follow smooth paths in layers.
  • Pressure energy: Energy due to the pressure applied by a fluid particle.
  • Kinetic energy: Energy due to the motion of fluid particles.
By understanding fluid dynamics, we can solve complex problems related to fluid behavior, such as determining wind speed between buildings.
Pressure Variation
Pressure variation plays a significant role in the application of Bernoulli's equation. Pressure is the force exerted by the fluid per unit area. In fluid dynamics, when a fluid flows from a wider to a narrower region, its speed increases, resulting in a lower pressure.
This principle is essential because Bernoulli鈥檚 equation assumes that the flow is incompressible, meaning density remains constant while the pressure can vary. For our exercise, the problem specifies that the pressure should not vary beyond 10%. This range ensures the equation holds without major density fluctuations.
Understanding pressure variation is crucial in fluid dynamics, especially when analyzing environments where pressure changes could impact flow, such as wind flowing between buildings. These variations help us predict and control the behavior in engineering projects or weather forecasting.
Air Density
Air density is a crucial factor in applying the incompressible Bernoulli equation. It indicates the mass of air per unit volume, typically measured in kg/m鲁. In standard conditions, air density is approximately 1.225 kg/m鲁.
For Bernoulli鈥檚 equation to apply, the problem assumes that air density doesn't change significantly. In practical terms, this means that pressure variations remain within a 10% threshold, allowing us to treat the fluid as incompressible at a constant density. While air density can influence how wind behaves, in many engineering applications, assuming a constant density simplifies calculations significantly.
This assumption allows for easier computation when analyzing fluid behavior, such as calculating the speed of wind between buildings, which in turn affects structural design and safety protocols.
Wind Speed Analysis
Wind speed analysis involves understanding how and why wind speeds vary across different environments. It often requires the use of fluid dynamics principles to predict changes in velocity due to obstacles or environmental factors.
In this exercise, we analyze wind speed as it increases when passing between buildings. When wind hits constricted spaces, it speeds up, which can be demonstrated using Bernoulli's equation. For instance, knowing the initial wind speed and air density, we use the equation to find that the maximum speed between the buildings could reach approximately 52.15 m/s, depending on the pressure drop.
This analysis is vital for assessing risks during severe weather events, like hurricanes, where wind speeds can increase dramatically in built environments. Understanding these speeds aids in urban planning and engineering, ensuring structures can withstand expected wind pressures.

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Most popular questions from this chapter

A steady, incompressible, two-dimensional velocity field is given by the following components in the \(x y\) plane: $$ v_{x}=2+1.5 x+0.75 y, \quad v_{y}=1+3 x+1 y $$ What is the acceleration of the fluid at \((x, y)=(2,4) ?\)

Experiments indicate that the shear stress, \(\tau_{0}\), on the wall of a 200 -mm pipe can be related to the flow in the pipe using the relation $$ \tau_{0}=0.04 \rho V^{2} $$ where \(\rho\) and \(V\) are the density and velocity, respectively, of the fluid in the pipe. If water at \(20^{\circ} \mathrm{C}\) flows in the pipe at a flow rate of \(60 \mathrm{~L} / \mathrm{s}\) and the pipe is horizontal, estimate the pressure drop per unit length along the pipe.

Water at \(25^{\circ} \mathrm{C}\) flows through a \(1.5-\mathrm{m}\) section of a converging conduit in which the velocity, \(V\) (in \(\mathrm{m} / \mathrm{s}\) ), in the conduit increases linearly with distance, \(x\) (in \(\mathrm{m}\) ), from the entrance of the section according to the relation $$ V=1.5(1+0.5 x) \mathrm{m} / \mathrm{s} $$ The flow can be approximated as being inviscid. (a) Determine the pressure gradient as a function of \(x\) and determine the pressure difference across the \(1.5-\mathrm{m}\) section of conduit by integrating the pressure gradient. (b) Determine the pressure difference across the conduit using the Bernoulli equation and compare your result with that obtained in part (a).

An airplane operates at a low elevation where the pressure is \(101 \mathrm{kPa}\) and the temperature is \(20^{\circ} \mathrm{C}\). (a) Neglecting compressibility effects, estimate the flight speed at which the stagnation pressure is \(15 \%\) higher than the static pressure. (b) Estimate (a) would differ if compressibility effects were taken by how much the result in part into account.

A 2 -m-long tank with a trapezoidal cross section is drained from the bottom by a rectangular (slot) opening in the bottom center of the tank. The drain slot extends over the entire \(2-\mathrm{m}\) length of the tank and is \(1 \mathrm{~cm}\) wide. The slot opening has been smoothed such that it has a discharge coefficient approximately equal to unity. The trapezoidal cross section of the tank has a bottom width of \(1 \mathrm{~m}\) and side slopes that are at \(60^{\circ}\) to the horizontal. If the initial depth of liquid in the tank is \(1 \mathrm{~m}\), how long does it take to drain the tank completely? Comment on the influence of the density of the liquid on the drain time.
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