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

AI Thread Summary
To calculate the height of mercury in a U-tube using Bernoulli's equation, start by determining the fluid velocities from the given flow rate and cross-sectional areas of the tube. Apply Bernoulli's equation to find the pressure difference between the two sides of the tube, which can be expressed as P1 - P2. This pressure difference can then be related to the height difference in the fluid columns using the equation P = ρgh. The final calculated height of mercury in the right side of the U-tube is approximately 5.82 cm. This method effectively combines fluid dynamics principles with practical measurements.
sarahmf
Messages
3
Reaction score
0

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 :)
 
Physics news on Phys.org
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.
 
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 :)
 
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.
 
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 :)
 
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
Back
Top