Siphon tube

A pipe of diameter D = 2 cm D = 2 \text{ cm} is connected to an open tank of large cross-section area containing a non-viscous fluid which flows in a streamlined manner. The level of fluid in the tank from the base is maintained at H = 50 cm H = 50 \ \text{cm} by some external means. A nozzle of diameter d = 1 cm d = 1 \text{ cm} fitted at the end of the pipe discharges water into the atmosphere.

Given that the density of the fluid is ρ = 2.5 g/cc \rho = 2.5 \text{ g/cc} , find the pressure ( ( to the nearest integer value in mm of Hg ) \text{mm of Hg}) at point C C which is at a height of h = 5 cm h = 5 \text{ cm} above the level of fluid in the tank.

Details and Assumptions:

  • Atmospheric pressure P 0 = 101325 Pa = 760 mm of Hg . P_0 = 101325 \text{ Pa} = 760 \text{ mm of Hg}.
  • Acceleration due to gravity g = 9.8 m/s 2 . g = 9.8 \text{ m/s}^2.


The answer is 745.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Tapas Mazumdar
Jan 22, 2018

Taking the mouth of the nozzle as the reference for zero gravitational potential and applying Bernoulli's theorem for the mouth of nozzle and the water in the tank, we get

P 0 + H ρ g + 0 = P 0 + 0 + 1 2 ρ v 2 v = 2 g H \begin{aligned} & P_0 + H \rho g + 0 = P_0 + 0 + \dfrac 12 \rho v^2 \\ \implies & v = \sqrt{2gH} \end{aligned}

Via equation of continuity we have

A V = a v π D 2 4 V = π d 2 4 2 g H V = d 2 D 2 2 g H \begin{aligned} & AV = av \\ \implies & \dfrac{\pi D^2}{4} V = \dfrac{\pi d^2}{4} \cdot \sqrt{2gH} \\ \implies & V = \dfrac{d^2}{D^2} \cdot \sqrt{2gH} \end{aligned}

Again taking mouth of nozzle as the reference for zero gravitational potential and applying Bernoulli's theorem for the mouth of nozzle and point C, we get

P C + ( H + h ) ρ g + 1 2 ρ V 2 = P 0 + 0 + 1 2 ρ v 2 P C = P 0 ( H + h ) ρ g + 1 2 ρ ( v 2 V 2 ) = P 0 ( H + h ) ρ g + 1 2 ρ ( 1 d 4 D 4 ) 2 g H = P 0 ρ g ( h + H d 4 D 4 ) \begin{aligned} P_C + (H+h) \rho g + \dfrac 12 \rho V^2 &=& P_0 + 0 + \dfrac 12 \rho v^2 \\ \implies P_C &=& P_0 - (H+h) \rho g + \dfrac 12 \rho \left( v^2 - V^2 \right) \\ &=& P_0 - (H+h) \rho g + \dfrac 12 \rho \left( 1 - \dfrac{d^4}{D^4} \right) 2gH \\ &=& P_0 - \rho g \left( h + H \cdot \dfrac{d^4}{D^4} \right) \end{aligned}

Putting required values, we get

P C = 99334.375 Pa = 745.069 mm of Hg 745 P_C = 99334.375 \ \text{Pa} = 745.069 \ \text{mm of Hg} \approx \boxed{745}

Couldn't we directly do P(atmosphere) = P( at C ) - density x h x g ,
as velocity of the liquid through out the siphon remains the same

Aswin Ramesh - 3 years, 4 months ago

Log in to reply

The level of water is maintained in the tank which further is characterised to have a large cross-section area. Thus, the speed with which water level is theoretically observed to be coming down is almost negligible and not the same as that in the siphon tube.

Moreover, the velocity of efflux is different than that of the pipe because it has a different cross-sectional area.

Tapas Mazumdar - 3 years, 4 months ago

Log in to reply

Does area of cross section even matter ?? Because according to Bernoulie's equation,ie. by conservation of energy, the velocity should be same through out the siphon, (as you said this velocity will be different from the rate at which the height column of the cylinder decreases) , shouldn't it ?

Aswin Ramesh - 3 years, 4 months ago

Log in to reply

@Aswin Ramesh Yes, the area of cross-section does matter. Since equal mass of water should through all points at equal intervals of time, for two different cross-sections, we should have

Mass of water passing through a given cross section = density × volume M = ρ ( A s ) = ρ ( A v t ) \text{Mass of water passing through a given cross section} = \text{density} \times \text{volume} \\ \implies M = \rho (A \cdot s) = \rho (A \cdot v \cdot t)

thus

ρ A 1 v 1 t = ρ A 2 v 2 t A 1 v 1 = A 2 v 2 = Φ \rho A_1 v_1 t = \rho A_2 v_2 t \implies A_1 v_1 = A_2 v_2 = \Phi

Thus, the volumetric flow rate Φ \Phi should be same and the velocities at different cross sections have to be different.

Tapas Mazumdar - 3 years, 4 months ago

Log in to reply

@Tapas Mazumdar Oh, i used to think that the velocity remains constant . Thanks for the help , but when i do solve the question using my method, I'm getting the correct answer. Is is a mere coincidence ?? I'm feeling is isn't.

Aswin Ramesh - 3 years, 4 months ago

Log in to reply

@Aswin Ramesh Can you show me your method?

Tapas Mazumdar - 3 years, 4 months ago

Log in to reply

@Tapas Mazumdar My bad, i did a calculation mistake :P

Aswin Ramesh - 3 years, 4 months ago

Is that a typo in the 1st line? That H

Md Junaid - 3 years, 3 months ago

Log in to reply

Thanks that was a big error. But the answer is still the same. Thanks for mentioning. :)

Tapas Mazumdar - 3 years, 3 months ago

Hey I got 743 . And rhen tried 745. Do you think am correct?

Md Zuhair - 3 years, 3 months ago

Log in to reply

You should've got 744.7. I have already given all the values needed.

Tapas Mazumdar - 3 years, 3 months ago

Log in to reply

Yeah ... Actually, I took g=10 I guess... and some other small errors :D.

Md Zuhair - 3 years, 3 months ago

Log in to reply

@Md Zuhair Yes. Assuming you took g=10, I said it would have been 744.7. Nevermind the answer is correct. :)

Tapas Mazumdar - 3 years, 3 months ago

Log in to reply

@Tapas Mazumdar Yeah. Small errors come, Btw, You deleted Whatsapp?

Md Zuhair - 3 years, 3 months ago

Log in to reply

@Md Zuhair Yes for JEE preparations

Tapas Mazumdar - 3 years, 3 months ago

Log in to reply

@Tapas Mazumdar I guess... you will get a nice IIT CLLGE

Md Zuhair - 3 years, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...