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Chapter 6: Problem 82

Water flows at low speed through a circular tube with inside diameter of 2 in. A smoothly contoured body of 1.5 in. diameter is held in the end of the tube where the water discharges to atmosphere. Neglect frictional effects and assume uniform velocity profiles at each section. Determine the pressure measured by the gage and the force required to hold the body.

Short Answer

Expert verified
The pressure measured by the gage is 0, and no force is required to hold the body.

Step by step solution

01

Determine the Areas

First, we will calculate the areas of the tube and the body using the formula for the the area of a circle, \(A=\pi r^2\), where \(r\) is the radius, which is half of the diameter. So, the area of the tube is \(A_t=\pi (1)^2= \pi\) sq.in., and the area of the body is \(A_b=\pi (0.75)^2= 0.5625 \pi\) sq.in.
02

Apply Bernoulli's Equation

Bernoulli's equation is given by \(P + 1/2 蟻v^2 + 蟻gh = constant\), where \(P\) is the pressure, \(蟻\) is the fluid density, \(v\) is the velocity and \(g\) is the acceleration due to gravity. Since the heights and velocities at both points (inside the tube and at the point where the water discharges to atmosphere) are the same, and we are neglecting frictional effect, the equation simplifies to \(P1 = P2\) where \(P1\) is the inside pressure and \(P2\) is the atmosphere pressure.
03

Calculate the Pressure Drop

From step 2 we have found that the pressure inside the tube is equal to atmospheric pressure. Without knowing the absolute value of this pressure, we cannot determine the force on the body. However, we know that any pressure difference will be relieved as water is discharged, without affecting the physical situation. Thus, the pressure \(P1\) or \(P2\) is 0.
04

Calculate the Force Required to Hold the Body

The force \(F\) required to hold the body is given by \(F = P 脳 A\), where \(P\) is the pressure and \(A\) is the area. Substituting \(P = 0\) from Step 3 and \(A = \pi\) sq.in. from Step 1, we get \(F = 0 脳 \pi = 0\) lb. Therefore, no force is required to hold the body.

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Most popular questions from this chapter

The flow field for a plane source at a distance \(h\) above an infinite wall aligned along the \(x\) axis is given by \\[ \begin{aligned} \vec{V} &=\frac{q}{2 \pi\left[x^{2}+(y-h)^{2}\right]}[x \hat{i}+(y-h) \hat{j}] \\\ &+\frac{q}{2 \pi\left[x^{2}+(y+h)^{2}\right]}[x \hat{i}+(y+h) \hat{j}] \end{aligned} \\] where \(q\) is the strength of the source. The flow is irrotational and incompressible. Derive the stream function and velocity potential. By choosing suitable values for \(q\) and \(h,\) plot the streamlines and lines of constant velocity potential.

The velocity distribution in a two-dimensional steady flow field in the \(x y\) plane is \(\vec{V}=(A x-B) \hat{i}+(C-A y) \hat{j}\) where \(A=2 \mathrm{s}^{-1}, B=5 \mathrm{m} \cdot \mathrm{s}^{-1},\) and \(\mathrm{C}=3 \mathrm{m} \cdot \mathrm{s}^{-1} ;\) the coordinates are measured in meters, and the body force distribution is \(\vec{g}=-g \hat{k} .\) Does the velocity field represent the flow of an incompressible fluid? Find the stagnation point of the flow field. Obtain an expression for the pressure gradient in the flow field. Evaluate the difference in pressure between point \((x, y)=(1,3)\) and the origin, if the density is \(1.2 \mathrm{kg} / \mathrm{m}^{3}\)

The velocity distribution in a two-dimensional, steady, inviscid flow field in the \(x y \quad\) plane is \(\vec{V}=(A x+B) \hat{i}+(C-A y) \hat{j}, \quad\) where \(A=3 \quad \mathrm{s}^{-1}, \quad B=6 \mathrm{m} / \mathrm{s}\) \(C=4 \mathrm{m} / \mathrm{s},\) and the coordinates are measured in meters. The body force distribution is \(\vec{B}=-g \hat{k}\) and the density is \(825 \mathrm{kg} / \mathrm{m}^{3} .\) Does this represent a possible incompressible flow field? Plot a few streamlines in the upper half plane. Find the stagnation point(s) of the flow field. Is the flow irrotational? If \(\mathrm{so},\) obtain the potential function. Evaluate the pressure difference between the origin and point \( (x, y, z)=(2,2,2) \)

Calculate the dynamic pressure that corresponds to a speed of $100 \mathrm{km} / \mathrm{hr}$ in standard air. Express your answer in millimeters of water.

A rectangular computer chip floats on a thin layer of air, \(h=0.5 \mathrm{mm}\) thick, above a porous surface. The chip width is \(b=40 \mathrm{mm},\) as shown. Its length, \(L,\) is very long in the direction perpendicular to the diagram. There is no flow in the \(z\) direction. Assume flow in the \(x\) direction in the gap under the chip is uniform. Flow is incompressible, and frictional effects may be neglected. Use a suitably chosen control volume to show that \(U(x)=q x / h\) in the gap. Find a general expression for the \((2 \mathrm{D})\) acceleration of a fluid particle in the gap in terms of \(q, h, x,\) and \(y\) Obtain an expression for the pressure gradient \(\partial p / \partial x\). Assuming atmospheric pressure on the chip upper surface, find an expression for the net pressure force on the chip; is it directed upward or downward? Explain. Find the required flow rate \(q\) \(\left(\mathrm{m}^{3} / \mathrm{s} / \mathrm{m}^{2}\right)\) and the maximum velocity, if the mass per unit length of the chip is \(0.005 \mathrm{kg} / \mathrm{m} .\) Plot the pressure distribution as part of your explanation of the direction of the net force.
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