Skip to main content

Pressures in rotating sealed U-tube

Short answer

Gauge pressures are p_C = 29.4 kPa, p_B = −29.8 kPa, p_A = −59.2 kPa, obtained by summing hydrostatic and centrifugal terms from the open limb D to sealed limb A.

The U-tube AB and CD shown in figure below filled with water. The tube AB is sealed where as tube CD is open to atmosphere. Find the pressure intensities at the points A, B and C where it is rotating with axis Y-Y with uniform rotation of 60 rpm. [Figure: U-tube with sealed limb AB (height 3 m) at 1 m from the axis Y-Y and open limb CD at 2 m on the other side of the axis]

Reading of the figure: the U-tube is filled with water; limb AB (sealed at the top A, height 3 m) is 1 m from the axis Y-Y; limb CD (open at the top D) is 2 m on the other side. Take B and C at the bottom, joined by a horizontal tube through the axis, and D at the same level as A (3 m above C). N=60N=60 rpm so ω=2π=6.283\omega=2\pi=6.283 rad/s. Pressures are gauge.

 A(sealed)            D(open)
  |                     |
  | 3 m                 | 3 m
  |                     |
 B+----------Y----------+C
  1 m                2 m

At rest the open end D is at atmospheric pressure. During rotation, pressure follows p=12ρω2r2−ρgz+constantp=\tfrac12\rho\omega^{2}r^{2}-\rho g z+\text{constant}. At D (r=2r=2, z=3z=3 above the base), p=0p=0.

At C (bottom of the open limb, r=2r=2 m)

Hydrostatic in the vertical limb (no change with rr along a vertical line):

pC=ρg(3)=1000(9.81)(3)=29.43 kPap_C=\rho g(3)=1000(9.81)(3)=29.43\ \text{kPa}

At B (bottom of the sealed limb, r=1r=1 m, same level as C)

pB=pC+12ρω2(rB2−rC2)p_B=p_C+\tfrac{1}{2}\rho\omega^{2}\left(r_B^{2}-r_C^{2}\right) pB=29.43+12(1000)(6.283)2(1−4)=29.43−59.22=−29.79 kPap_B=29.43+\tfrac12(1000)(6.283)^{2}(1-4)=29.43-59.22=-29.79\ \text{kPa}

At A (top of the sealed limb)

pA=pB−ρg(3)=−29.79−29.43=−59.22 kPap_A=p_B-\rho g(3)=-29.79-29.43=-59.22\ \text{kPa}

Answer: pA≈−59.2p_A\approx-59.2 kPa, pB≈−29.8p_B\approx-29.8 kPa, pC≈+29.4p_C\approx+29.4 kPa (gauge). The absolute pressure at A is still about 42 kPa, above the vapour pressure of water, so the column does not break.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Fluid Mechanics chapter 2 questions →