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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

A horizontal axisymmetric jet of air with 0.4 in. diameter strikes a stationary vertical disk of 7.5 in. diameter. The jet speed is \(225 \mathrm{ft} / \mathrm{s}\) at the nozzle exit. A manometer is connected to the center of the disk. Calculate (a) the deflection, if the manometer liquid has \(\mathrm{SG}=1.75,(\mathrm{b})\) the force exerted by the jet on the disk, and (c) the force exerted on the disk if it is assumed that the stagnation pressure acts on the entire forward surface of the disk. Sketch the streamline pattern and plot the distribution of pressure on the face of the disk.

Steady, frictionless, and incompressible flow from left to right over a stationary circular cylinder, of radius \(a,\) is represented by the velocity field \\[ \vec{V}=U\left[1-\left(\frac{a}{r}\right)^{2}\right] \cos \theta \hat{e}_{r}-U\left[1+\left(\frac{a}{r}\right)^{2}\right] \sin \theta \hat{e}_{\theta} \\] Obtain an expression for the pressure distribution along the streamline forming the cylinder surface, \(r=a\). Determine the locations where the static pressure on the cylinder is equal to the freestream static pressure.

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.

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