How to Calculate Fluid Height in a U-Tube Using Bernoulli's Equation

[sarahmf]

Homework Statement

The following tube has dimensions d1=1.13 cm and d2=3.60 mm. Air (density = 1.28 kg/m3) flows through the tube at a rate of 1131.0 cm3/s. Assume that air is an ideal fluid. What is the height h of mercury (density = 13600.0 kg/m3) in the right side of the U tube?

Homework Equations

Bernoulli's equation:
P1+.5rho*v1^2 + rho*g*h1=P2+.5rho*v2^2 + rho*g*h2

A1V1=A2V2

The Attempt at a Solution

I'm really not sure where to start here. I wanted to use Bernoulli's equation but I don't have the velocity or pressure. Also, I'm not sure how to convert flow rate to velocity. Any help would be appreciated, it's due in about 24 hours :)

 

[edgepflow]

OK, first try to determine an equation for volumetric flow rate (given) as a function of flow cross section area (find from given diameter) and fluid velocity.

Next, try to apply Bernoulli's equation you listed to find the difference in static pressure:
P1 - P2.

Finally, use an equation that relates fluid static pressure to fluid column height. This will be the height or "deflection" of the U-tube manometer.

 

[sarahmf]

Alright so

Flow=A*V
V1=Flow/A1 =1131.0/(pi*(1.13/2)^2)
V1=1126 cm/s
V2=Flow/A2=1131.0/(pi*(0.36/2)^2)
V2=34907 cm/s

P1-P2=.5rho*v2^2 + rho*g*h2 - .5rho*v1^2 - rho*g*h1
P1-P2=.5*1.28 kg/m^3*1m^3/1000000 cm^3 (is my conversion right) * 34907^2 + 1.28/1000000*9.8*h2 - .5*1.28/1000000*1126^2 - 1.28/1000000*9.8*h1

and is the equation i have to use
P2=P1+rho*g*h ?

I'm not sure if this is right, feedback would be appreciated :)

 

[edgepflow]

Yes, you are on the right track. You can either solve for h2 - h1 in your Bernoulli equation. Or look at it from the point of view of manometer measurement: figure out P2 - P1 due to velocity changes only and then apply your P1 - P1 = rho*g*h to find h.

 

[sarahmf]

Alright so if I solve for P2-P1=rho*g*h I get 0.0582 m=5.82 cm, which is recognized as right..thank you so much :)