--- title: "Measuring pressure difference in a pipe" subject: "Hydro Power" chapter: 1 key: "semester_6/hydro_power:measuring-pressure-difference" source: https://www.bctengineer.com/q/semester_6/hydro_power/measuring-pressure-difference site: BCT Engineering --- # Measuring pressure difference in a pipe Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics - Asked in: 2070 Magh, 6 marks - Video: [Pressure Measurement Manometers](https://www.youtube.com/watch?v=oXXGIFSmZRE) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/hydro_power/measuring-pressure-difference - Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1 ## Question How do you measure difference of pressure between any two points in a pipe? Explain with neat sketch and mathematical expression. ## Short answer A differential manometer uses a U-tube with heavy liquid. For points at different levels: pA − pB = ρ_g g h + ρ_2 g h_B − ρ_1 g h_A. If in the same horizontal pipe with the same liquid: pA − pB = gh(ρ_g − ρ). ## Answer The pressure difference between two points in a pipe (or between two pipes) is measured with a **differential manometer**: a U-tube containing a heavy manometric liquid (usually mercury), with its two limbs connected to the two points. #### U-tube differential manometer (points at different levels) ```text A o | pipe liquid S1 o B | | pipe liquid S2 | h_A | h_B | h | X --+-----. .-------------+ | Hg | | Hg surface \____|__|____/ mercury (S_g) ``` Let A be $h_A$ above the datum X–X (the lower mercury surface), B be $h_B$ above the higher mercury surface, and $h$ the difference between the mercury levels. Pressure at X–X in the left limb $= p_A + \rho_1 g h_A$. Pressure at X–X in the right limb $= p_B + \rho_2 g h_B + \rho_g g h$. $$ p_A - p_B = \rho_g g h + \rho_2 g h_B - \rho_1 g h_A $$ #### Both points at the same level, same liquid If A and B are in the same horizontal pipe carrying a liquid of density $\rho$: $$ p_A - p_B = g h(\rho_g - \rho) $$ For water and mercury: $p_A - p_B = 12.6\,\rho_w g h$. #### Inverted U-tube manometer For small pressure differences, a light liquid (air or oil) is used at the top of an inverted U-tube: $$ p_A - p_B = \rho_1 g h_1 - \rho_2 g h_2 - \rho_s g h $$ where $h_1$ and $h_2$ are the depths of A and B below their interfaces with the light liquid, $h$ is the level difference between the two interfaces, and $\rho_s$ is the density of the light liquid. #### Steps of measurement 1. Connect each limb to its pipe tapping, filling the leads with the pipe liquid (no air bubbles). 2. Let the manometer settle and read the level difference $h$. 3. Choose a datum at the lower interface and equate the pressures in both limbs. Other devices: a pair of Bourdon gauges, or electronic differential pressure transducers, for large differences. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The pressure difference between two points in a pipe is measured with: A. A Bourdon gauge only B. A differential manometer, a U-tube with heavy liquid such as mercury C. A piezometer on one point D. A pitot tube Answer: B. A differential manometer connects the two points to the limbs of a U-tube. (Key point: Instrument) 2. For two points at the same level in a pipe carrying liquid of density rho, with mercury reading h: A. p_A - p_B = g h (rho_g - rho) B. p_A - p_B = g h (rho_g + rho) C. p_A - p_B = rho g h D. p_A - p_B = g h rho_g / rho Answer: A. The pipe liquid in the manometer leads cancels part of the mercury head, so rho_g - rho is used. (Key point: Same-level formula) 3. For water in the pipe and mercury in the manometer, p_A - p_B equals: A. 13.6 rho_w g h B. 12.6 rho_w g h C. 14.6 rho_w g h D. rho_w g h Answer: B. (13.6 - 1) = 12.6, so p_A - p_B = 12.6 rho_w g h. (Key point: Water and mercury) 4. An inverted U-tube with air or light oil at the top is used for: A. Large pressure differences B. Small pressure differences C. Measuring atmospheric pressure D. Measuring temperature Answer: B. A light fluid gives a large reading for small pressure differences, where mercury would give too small a reading. 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