Derive the Venturi Meter eqn from the Bernoulli eqn

In summary, the conversation discusses a problem with equations and substitutions involving pressure and fluid dynamics. The person is seeking help in understanding where they went wrong in their calculations and how to properly express the equations in terms of various variables. The introduction of the height variable helps clarify the physical sense behind the equations.
  • #1

GreyNoise

Gold Member
32
6
Homework Statement
By applying Bernoulli's equation and the equation of continuity to points 1 and 2 of Fig. 16-14 [see attached file], show that the speed of the flow at the entrance is
v1 = a*sqrt{(2(dens' - dens)gh)/(dens(A^2-a^2))}
Relevant Equations
0.5*dens*v_1^2 + p_1 = 0.5*dens*v_2^2 + p_2 Bernoulii eqn
A*v_1 = a*v_2 continuity eqn
Advanced apologies for this format; I am posting my question as an the image b/c the Latex is being very buggy with me, and I lost a kind of lengthy post to it. Can anyone show me what I am doing wrong? I have attached a pdf version for easier reading if need be.
pr-43-p-290-h-r-text-ed.jpg
 

Attachments

  • pr-43-p-290-h-r-text-ed.pdf
    102.1 KB · Views: 62
Physics news on Phys.org
  • #2
GreyNoise said:
1657147269864.png


From equation (1) you can see that ##p_2## must be less than ##p_1## because ##v_2 > v_1## (from the continuity equation). So, ##p_2 < p_1##.

However, in equation (2) you let ##p_1 = \rho g h## and ##p_2 =\rho' \, gh##.
But ##\rho' \, > \rho##. So, these substitutions would imply that ##p_2 > p_1##, which contradicts ##p_2 < p_1##. So, letting ##p_1 = \rho g h## and ##p_2 =\rho' \, gh## can't be correct.

Assume we can take points 1 and 2 to be at the same horizontal level:
1657148480656.png


Introduce the height ##H## as shown. Can you express ##p_1## in terms of ##p_c## , ##\rho##, ##g##, ##H##, and ##h##? Likewise, can you relate ##p_2## and ##p_d##?
 
  • #3
Thnx so much for the response TNsy. Pointing out my contradiction between lines (1) and (2) was the big aha moment for me, and including the ##\rho gH## term makes the physical sense clear now.
 
  • Like
Likes TSny

Suggested for: Derive the Venturi Meter eqn from the Bernoulli eqn

Back
Top