Solving a Water Pressure Problem: Gauge Pressure at B

[Coop]

Hi,

I am working on this water pressure problem,

given:

v_A = 2.0 $\frac{m}{s}$
gauge pressure_A = 50 kPa
the view is from above, no height changes

find: gauge pressure @ B

so,

$A_Av_A=2(A_Bv_B)$
$1.5*10^{-2}m^2(2.0\frac{m}{s})=2(5*10^{-3}m^2v_B)$
$v_B=9.0\frac{m}{s}$

Bernoulli's equation (w/o the $\rho gh$ components bc height is constant):

$p_A+\frac{1}{2}\rho _Av_A^2=p_B+\frac{1}{2}\rho _Bv_B^2$

My question is, how come the right side of the equation is not $2(p_B+\frac{1}{2}\rho _Bv_B^2)$? Shouldn't there be a coefficient of two, because the pipe splits in half?

Thanks

 

[dauto]

1st: I think you made a mistake in the calculation of v_B

2nd: No there is no factor of 2. why should there be one? Bernoulli equation is just the energy conservation for some parcel of water as if follows its path along the tubes. Side branches have no effect.

 

[Coop]

Side branches have no effect.

Thanks

And what mistake do you think I made for v_B? It worked out giving me the right answer .

 

[dauto]

I got 3/s from the middle equation. May be the middle equation is wrong and the final result is correct. Hard to tell since you never specified the shape of the ducts (circular cross section I assume, but is it?).

 

[Coop]

I got 3/s from the middle equation. May be the middle equation is wrong and the final result is correct. Hard to tell since you never specified the shape of the ducts (circular cross section I assume, but is it?).

Oh you're right, sorry. Yeah they are circular pipes but I just wrote down the area incorrectly. I wrote down the pipes' radii for their area when I shouldn't wrote their radii^2*pi. Thanks for the help!

 

[sophiecentaur]

As long as the pipe cross sections are similar figures, the shape doesn't matter (assuming no turbulence) as it's only the ratio of the areas that counts.
The factor of two, because of two pipes, has been 'used' in the calculation of the output velocity.
Pressure and velocity are intensive variables so the number of pipes doesn't matter, once you've calculated the velocity.

 

[pscience]

in bernoulli's equation P+ρ g h =constant , are P and ρ g h different ? aren't they the same? thanks.

 

[mfb]

In general, they are different.
In particular, h depends on the arbitrary definition of zero height. You can choose whatever you like, as only height differences have a physical relevance.