# Bernoulli’s Equation and External Forces

• Khalid Qasim
In summary, Bernoulli's equation and external forces are used to calculate the velocities and momentum of a fluid being sprayed from a nozzle.
Khalid Qasim
Thread moved from the technical forums
Summary:: Bernoulli’s Equation and External Forces

Hello all, I am a bit confused on this question:

Water flows through a spray nozzle (as shown in the figure below) at a rate of 50 cm3 h -1 .
The internal inlet diameter of the nozzle entrance is 5 cm and the internal exit diameter of the nozzle is 1 cm.
The gauge pressure is 2.75 bar at the entrance of the nozzle and the pressure outside the nozzle exit is atmospheric.
Calculate the external force required to keep the spray nozzle in place, and the direction of this force.
State any assumptions you make and give all the steps in your calculation.

I have tried using bernoulli's eq to find different velocities to calculate momentums to hopefully find a thrust force, but to no avail. Can someone help?

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Please show us what you tried. Then perhaps we can diagnose why "to no avail."

kuruman said:
Please show us what you tried. Then perhaps we can diagnose why "to no avail."
I've crossed it out now, but what I had done:
Found force at the entrance by multiplying 3.76 bar and area at start.
Found force at the exit by multipying 1.01bar and area at exit.

Now for the next part:
Finding the velocities I did: Start: (50x(10^-2)^3)/(3600x(area at start)) = 0.0000708 (which already seems wrong)
Now subbing into bernoullis eq i got the exit velocity as 23.45
so i tried to do a momentum change, but was unsure of the time interval to even use, and I got a bit too confused to continue

What is your strategy for finding the external force required to keep the spray nozzle in place? To guide your thinking, consider two extreme situations in which the pressure at the entrance is 2.75 bar: (a) the exit diameter is plugged and there is no water flow; (b) the exit diameter is the same as the entrance diameter and the fluid flows freely. How would you calculate the required force in each case?

When you respond, please use equations in symbolic form, not numbers and especially not numbers without units attached to them.

berkeman
My first question for you is "Is 2.75 bars at the entrance to the nozzle together with the inlet and exit velocities and the exit pressure of 1 bar consistent with the Bernoulli equation?"

My second question is whether you are familiar with the use of a macroscopic momentum balance to determine the force exerted by the nozzle on the fluid within the nozzle?

## 1. What is Bernoulli's equation?

Bernoulli's equation is a fundamental principle in fluid dynamics that relates the pressure, velocity, and elevation of a fluid in a steady flow. It states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid remains constant along a streamline.

## 2. How is Bernoulli's equation derived?

Bernoulli's equation is derived from the application of the conservation of energy principle to a fluid flow. It assumes that the fluid is incompressible, inviscid, and has a steady flow. The equation is derived by considering a small volume of fluid along a streamline and equating the change in total energy to the work done by external forces.

## 3. What are external forces in Bernoulli's equation?

External forces in Bernoulli's equation refer to any forces acting on the fluid that are not accounted for in the pressure, velocity, and elevation terms. These forces can include gravity, friction, and other external forces such as pumps or turbines.

## 4. How does Bernoulli's equation apply to real-world situations?

Bernoulli's equation has many applications in real-world situations, such as in the design of airplanes, cars, and other vehicles. It is also used in the design of pipes, pumps, and turbines for fluid transportation and energy production. Additionally, it is used in weather forecasting and understanding the behavior of fluids in natural phenomena such as ocean currents and air flow.

## 5. What are the limitations of Bernoulli's equation?

Bernoulli's equation has some limitations, such as assuming an ideal fluid with no viscosity or compressibility. It also assumes steady flow and neglects the effects of turbulence. In real-world situations, these assumptions may not hold true, and the equation may need to be modified or used in conjunction with other equations to accurately describe the fluid flow.

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