Derive the Venturi Meter eqn from the Bernoulli eqn

[GreyNoise]

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.

 

[TSny]

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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:

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 $p_d$?

 

[GreyNoise]

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.