Fluid mechanics: Bernoulli's equation and conservation of mass

In summary: Pa at the venturi throat.If you want to take into account the pressure distribution over the cross section - yes. But since you treat the problem as 1-dimensional to solve the Bernoulli equation you don't take it into account.
  • #1
TimeRip496
254
5

Homework Statement


A mechanical servo-mechanism comprising of a movable piston-cylinder within a vertical cylinder operates based on a venturi contraction in a horizontal 350mm diameter pipe that delivers a fluid of relative density 0.95. The upper end of the 100mm diameter vertical cylinder is connected by a pipe to the throat of the venturi, while the lower end of the cylinder is connected by a pipe to the pipe inlet. The piston in the cylinder is to be lifted vertically when the flow through the venturi exceeds 0.15m3 /s, such that it will activate a controller to throttle back the flow rate. Note that the piston rod is known to have a diameter of 20mm and that it passes through both ends of the vertical cylinder. Neglecting friction, calculate the diameter required of the venturi throat if the gross effective load on the piston rod is 180N.
Ans: 0.162m
upload_2018-8-24_16-8-57.png

Homework Equations


Benoulli equation & conservation of mass

The Attempt at a Solution


I tried solving using Bernoulli's equation and conservation of mass.

$$0.15=0.175^2\pi* v_p=r_t^2*\pi*v_t$$
where the subscript p represents location at pipe inlet while t represents location at throat

Area of cylinder is,$$(0.05^2-0.01^2)\pi = 0.00024\pi$$
Pressure difference in area of cylinder is,
$$P_p -P_t=180/(0.0024\pi)=23873$$
$$P_p + 0.5\rho v_p^2 = P_t + 0.5\rho v_t^2 +\rho *g(0.175-r_t)$$
$$\rho=950, \ \ g=9.81, \ \ v_p=1.56$$
$$r_t=\sqrt{\frac{0.15}{\pi*v_t}}$$
After subbing in, $$25028.96=475*v_t^2+9319.5(0.175-\sqrt{\frac{0.15}{\pi*v_t}})$$

I can only solved after I can get velocity at throat. But I am stuck at the above part.
 

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  • #2
Your post would be easier to read if you'd use symbols without plugging in preliminary numbers.

What you've achieved so far:

1) Mass conservation:

$$\dot{V}=r_p^2 \pi v_p = r_t^2 \pi v_t $$

2) The necessary pressure difference to generate the required force ##F_{Piston}##:

$$p_p-p_t=\left( r_{out}^2-r_{in}^2\right)\pi F_{Piston}=A_{Piston} F_{Piston} $$

TimeRip496 said:

The Attempt at a Solution


I tried solving using Bernoulli's equation and conservation of mass.

$$P_p + 0.5\rho v_p^2 = P_t + 0.5\rho v_t^2 +\rho *g(0.175-r_t)$$

3) Bernoulli:

$$p_p+\frac{v_p^2\rho}{2}=p_t+\frac{v_t^2\rho}{2}\underbrace{+\rho g \left(r_p-r_t\right)}_{?}$$

Why did you add this last term to the RHS of the Bernoulli equation?
 
  • #3
stockzahn said:
Your post would be easier to read if you'd use symbols without plugging in preliminary numbers.

What you've achieved so far:

1) Mass conservation:

$$\dot{V}=r_p^2 \pi v_p = r_t^2 \pi v_t $$

2) The necessary pressure difference to generate the required force ##F_{Piston}##:

$$p_p-p_t=\left( r_{out}^2-r_{in}^2\right)\pi F_{Piston}=A_{Piston} F_{Piston} $$
3) Bernoulli:

$$p_p+\frac{v_p^2\rho}{2}=p_t+\frac{v_t^2\rho}{2}\underbrace{+\rho g \left(r_p-r_t\right)}_{?}$$

Why did you add this last term to the RHS of the Bernoulli equation?
I think because the height difference for the tube at the throat compared to the tube at the pipe inlet will have gravitational energy which will affect the flow in the system. The rp is supposed to be on the LHS and I bring it to the RHS to include the gravitational effects due to height difference.
 
  • #4
TimeRip496 said:
I think because the height difference for the tube at the throat compared to the tube at the pipe inlet will have gravitational energy which will affect the flow in the system. The rp is supposed to be on the LHS and I bring it to the RHS to include the gravitational effects due to height difference.

Where is (the line of) the center of mass of the flow at the entrance (position ##p##) and where is it at the most narrow cross section (position ##t##)?

Edit: In the drawing it is marked very well...
 
  • #5
stockzahn said:
Where is (the line of) the center of mass of the flow at the entrance (position ##p##) and where is it at the most narrow cross section (position ##t##)?

Edit: In the drawing it is marked very well...
Thanks! However, in reality, it is supposed to include the gravitational effects right? This is because the tubes used to measure the pressure are not placed in the center!
 
  • #6
TimeRip496 said:
Thanks! However, in reality, it is supposed to include the gravitational effects right? This is because the tubes used to measure the pressure are not placed in the center!

If you want to take into account the pressure distribution over the cross section - yes. But since you treat the problem as 1-dimensional to solve the Bernoulli equation you don't take it into account. I suppose you could use the difference of the fluid height to correct the pressures at the small pipes connected with the piston:

$$p_p + \rho g 2 r_p - \left(p_t+\rho g 2 r_t\right)= p_p-p_t+\rho g 2 \left(r_p-r_t\right)=\left( r_{out}^2-r_{in}^2\right)\pi F_{Piston}=A_{Piston} F_{Piston} $$

This would lead to the same problem you had (a equation you have to solve numerically). However neglecting the hydrostatic head with the given geometry results in a mistake of about 10 % (a little less).

Edit: If there is no flow the difference in hydrostatic head exists as well, therefore it cancels out before and during the measurement. So maybe the problem statement should say: a force of 180 N compared to the fluid at rest.
 
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  • #7
stockzahn said:
If you want to take into account the pressure distribution over the cross section - yes. But since you treat the problem as 1-dimensional to solve the Bernoulli equation you don't take it into account. I suppose you could use the difference of the fluid height to correct the pressures at the small pipes connected with the piston:

$$p_p + \rho g 2 r_p - \left(p_t+\rho g 2 r_t\right)= p_p-p_t+\rho g 2 \left(r_p-r_t\right)=\left( r_{out}^2-r_{in}^2\right)\pi F_{Piston}=A_{Piston} F_{Piston} $$

This would lead to the same problem you had (a equation you have to solve numerically). However neglecting the hydrostatic head with the given geometry results in a mistake of about 10 % (a little less).

Edit: If there is no flow the difference in hydrostatic head exists as well, therefore it cancels out before and during the measurement. So maybe the problem statement should say: a force of 180 N compared to the fluid at rest.
Thanks a lot!
 
  • #8
stockzahn said:
If you want to take into account the pressure distribution over the cross section - yes. But since you treat the problem as 1-dimensional to solve the Bernoulli equation you don't take it into account. I suppose you could use the difference of the fluid height to correct the pressures at the small pipes connected with the piston:

$$p_p + \rho g 2 r_p - \left(p_t+\rho g 2 r_t\right)= p_p-p_t+\rho g 2 \left(r_p-r_t\right)=\left( r_{out}^2-r_{in}^2\right)\pi F_{Piston}=A_{Piston} F_{Piston} $$

This would lead to the same problem you had (a equation you have to solve numerically). However neglecting the hydrostatic head with the given geometry results in a mistake of about 10 % (a little less).

Edit: If there is no flow the difference in hydrostatic head exists as well, therefore it cancels out before and during the measurement. So maybe the problem statement should say: a force of 180 N compared to the fluid at rest.
The system is hydroststic in the vertical direction, so the vertical pressure variations cancel out.
 
  • #9
Chestermiller said:
The system is hydroststic in the vertical direction, so the vertical pressure variations cancel out.

I think that's what I tried to say with my last sentence:

stockzahn said:
Edit: If there is no flow the difference in hydrostatic head exists as well, therefore it cancels out before and during the measurement. So maybe the problem statement should say: a force of 180 N compared to the fluid at rest.

Or do you mean something else and I did get it wrong?
 
  • #10
I didn't quite follow, so I thought I would clarify a little more.
 

Related to Fluid mechanics: Bernoulli's equation and conservation of mass

1. What is Bernoulli's equation?

Bernoulli's equation is a fundamental principle in fluid mechanics that relates the pressure, velocity, and height of a fluid within a continuous flow. It states that as the speed of a fluid increases, the pressure within the fluid decreases.

2. How is Bernoulli's equation derived?

Bernoulli's equation is derived from the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or converted. In fluid mechanics, this translates to the idea that the total energy of a fluid remains constant as it flows. Bernoulli's equation is a simplified version of this principle, taking into account the specific conditions of a fluid flow.

3. What are some practical applications of Bernoulli's equation?

Bernoulli's equation has numerous practical applications in everyday life. Some common examples include the lift force on airplane wings, the flow of water through a pipe, and the operation of carburetors in cars. It is also used in various engineering and scientific fields, such as hydraulics, aerodynamics, and meteorology.

4. What is conservation of mass in fluid mechanics?

Conservation of mass is a fundamental principle in fluid mechanics that states that the mass of a fluid remains constant as it flows. This means that the mass of the fluid entering a system is equal to the mass of the fluid leaving the system, even if it undergoes changes in velocity or pressure.

5. How is conservation of mass related to Bernoulli's equation?

Bernoulli's equation and conservation of mass are both fundamental principles in fluid mechanics and are closely related. Bernoulli's equation takes into account the energy changes within a fluid flow, while conservation of mass ensures that the total mass within the flow remains constant. Together, these principles help us understand and predict the behavior of fluids in various scenarios.

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