Pressure and heights of a manometer -- Find the fluid flow rate

In summary, the flow rate of water can be found using the Bernoulli and continuity equations. However, there seems to be an error in the given solution, which may be due to a sign error in the last term of the second equation.
  • #1
Jenny Physics
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Homework Statement


Find the flow rate of water. The fluid in the manometer is mercury. ##d_{1},d_{2}## are the diameters in the figure
35k3bqr.png


Homework Equations


Bernoulli, continuity equations

The Attempt at a Solution



We know that

##p_{1}=p_{1Top}+\rho_{water}gh_{2}##, ##p_{2}=p_{2Top}+\rho_{water}gh_{3}##

which means

##p_{1Top}-p_{2Top}=p_{1}-p_{2}+\rho_{water}gh##

on the other hand

##p_{1}-p_{2}=\rho_{Hg}gh##

so

##p_{1Top}-p_{2Top}=gh(\rho_{water}+\rho_{Hg})##

But in the solution it is

##p_{1Top}-p_{2Top}=gh\rho_{Hg}##

where did I go wrong?
 

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  • #2
Jenny Physics said:
We know that

##p_{1}=p_{1Top}+\rho_{water}gh_{2}##, ##p_{2}=p_{2Top}+\rho_{water}gh_{3}##

which means

##p_{1Top}-p_{2Top}=p_{1}-p_{2}+\rho_{water}gh##
Check the sign of the last term on the right side of the second equation. This will modify your result.
Otherwise, I don't see anything wrong with your work. It does appear that the solution you were given is incorrect.
 
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Related to Pressure and heights of a manometer -- Find the fluid flow rate

1. What is a manometer and how does it measure pressure?

A manometer is a device used to measure pressure, particularly in gas or liquid systems. It works by comparing the pressure of the fluid being measured to a reference pressure, usually atmospheric pressure. This is done by using a U-shaped tube filled with a fluid, such as water, and measuring the difference in height between the two arms of the tube.

2. How does the height of a manometer relate to pressure?

The height of the manometer is directly proportional to the pressure being measured. This means that as the pressure increases, the height of the fluid in the tube will also increase. This relationship is described by the equation P = ρgh, where P is pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid in the tube.

3. How can the height of a manometer be used to calculate fluid flow rate?

The height of the manometer can be used to calculate fluid flow rate by using the Bernoulli's equation, which states that the total energy of a fluid remains constant throughout a system. By knowing the height difference between the two arms of the manometer and the density of the fluid, the flow rate can be determined.

4. What factors can affect the accuracy of a manometer measurement?

There are several factors that can affect the accuracy of a manometer measurement, including the type and density of the fluid being measured, the temperature and atmospheric pressure, and the precision of the instrument itself. It is important to carefully calibrate and monitor these factors to ensure accurate measurements.

5. How can the manometer be used in different industries?

The manometer has a wide range of applications in different industries, including HVAC systems, chemical and pharmaceutical processes, and fluid mechanics research. It is often used to monitor and control pressure in various systems, as well as to measure flow rates and detect leaks. Its versatility and accuracy make it a valuable tool in many fields of science and engineering.

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