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Manometer between oil and water pipes

Short answer

At D–D: left, p_A + 0.875×9.81×1.5 = p_A + 12.876 kN/m²; right, p_B + 1.25×9.81×2.25 = p_B + 27.591 kN/m². Equate: p_A − p_B = 14.715 kN/m², pipe A pressure higher.

Two pipes as shown in figure convey oil of specific gravity of 0.875 and water respectively. Both the liquids in the pipes are under pressure. The pipes are connected to a U-tube manometer and the hoses connecting the pipes to the tubes are filled with the corresponding liquids. Find the difference of pressure in two pipes if the level of manometric liquid having a specific gravity of 1.25 is 2.25 m higher in the right limb than the lower level of oil in the left limb of the manometer. [Figure: pipe A (oil) and pipe B (water) connected through an inverted U-tube manometer; the centre of A is 1.5 m above the lower oil level D–D, and the manometric liquid in the right limb stands 2.25 m above D–D (level E–F).]

Equate the pressures on the horizontal plane D–D (the lower oil level / top of the manometric liquid in the left limb), since below it the same manometric liquid is continuous.

Given: oil S1=0.875S_1 = 0.875, manometric liquid S=1.25S = 1.25, water S=1S = 1. Centre of A is 1.5 m above D–D. Manometric liquid in the right limb stands 2.25 m above D–D (level E–F). The centre of B is taken at level E–F, as no other height is given.

   A (oil) o                    o B (water) -- E-F
           |                    |
           | 1.5 m oil          | 2.25 m of
           |                    | liquid S=1.25
   D ------+---- S = 1.25 ------+------ D

Pressure at D–D, left limb

pA+0.875×9.81×1.5=pA+12.876 kN/m2p_A + 0.875 \times 9.81 \times 1.5 = p_A + 12.876\ \text{kN/m}^2

Pressure at D–D, right limb

pB+1.25×9.81×2.25=pB+27.591 kN/m2p_B + 1.25 \times 9.81 \times 2.25 = p_B + 27.591\ \text{kN/m}^2

Equate

pA+12.876=pB+27.591pA−pB=27.591−12.876=14.715 kN/m2\begin{aligned} p_A + 12.876 &= p_B + 27.591 \\ p_A - p_B &= 27.591 - 12.876 = 14.715\ \text{kN/m}^2 \end{aligned}

Answer: pA−pB=14.715p_A - p_B = 14.715 kN/m² (≈ 14.72 kPa); pipe A (oil) is at the higher pressure.

✓ Answer checked by Claude Opus 5.5 (high effort)

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