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Chapter 3: Problem 119

A centrifugal micromanometer can be used to create small and accurate differential pressures in air for precise measurement work. The device consists of a pair of parallel disks that rotate to develop a radial pressure difference. There is no flow between the disks. Obtain an expression for pressure difference in terms of rotation speed, radius, and air density. Evaluate the speed of rotation required to develop a differential pressure of \(8 \mu \mathrm{m}\) of water using a device with a \(50 \mathrm{mm}\) radius.

Short Answer

Expert verified
To obtain a solution, one would substitute the known values into the formula for the speed of rotation. The exact numerical result will depend on the specific density and pressure values used.

Step by step solution

01

Understand Bernoulli's Equation

Bernoulli's equation for an inviscid flow (flow without viscosity) of an incompressible fluid (where density is constant) is \[ P + 0.5*蟻*v^2 + 蟻*g*h = constant \] where P represents the pressure of the fluid, 蟻 is the density of the fluid, v is the flow velocity, g is the acceleration due to gravity, and h is the height from the reference plane.
02

Derive the Expression for Pressure Difference

When the disks rotate, a radial pressure difference 未P is generated. For small pressure differences where centrifugal forces dominate and gravitation can be ignoring, Bernoulli's equation simplifies to: \[ P + 0.5*蟻*v^2 = constant \] The pressure difference is then given by: \[ 未P = 0.5*蟻*(v2^2 - v1^2) \] Since there is no flow between the discs, we can let \( v1 = 0 \) and \( v2 = 蠅*r \), where 蠅 represents the rotation speed and r is the radius of the disc. This leads to the final expression: \[ 未P = 0.5*蟻*蠅^2*r^2 \] where 未P is differential pressure, 蟻 is the density of the air, 蠅 is the rotational speed, and r is the radius of the device.
03

Calculate the Speed of Rotation

Rearranging the previous expression to isolate for \( 蠅 \), the following formula is obtained: \[ 蠅 = sqrt((2*未P) / (蟻*r^2)) \] Given that the differential pressure 未P is \(8 \mu m\) of water, the radius r is 50 mm or 0.05 m, and the density of air 蟻 (at sea level and at room temperature) is approximately 1.2 kg/m^3, substitution yields the speed of rotation.

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Most popular questions from this chapter

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