Differential manometer Equation, Derivation & Rearrangements

Core idea

Overview

When the same process fluid fills both connection legs, the differential manometer simplifies to the density difference between the manometer fluid and the process fluid times gravity and height difference.

When to use: Use this simplified form for a U-tube differential manometer connected between two points in the same fluid.

Why it matters: The equation turns a visible column-height difference into a pressure difference across a pipe, filter, or restriction.

Symbols

Variables

$P_{1}$ - $P_{2}$ = Pressure Difference, $ρ_{m}$ = Manometer Fluid Density, $ρ$ = Process Fluid Density, g = Gravitational Acceleration, h = Height Difference

P_{1} - P_{2}

Pressure Difference

Pa

ρ_{m}

Manometer Fluid Density

k g / m^{3}

ρ

Process Fluid Density

k g / m^{3}

g

Gravitational Acceleration

m / s^{2}

h

Height Difference

m

Walkthrough

Derivation

Derivation of Differential manometer

The simplified differential manometer equation comes from the general pressure balance when both connection legs contain the same process fluid.

The two process-fluid legs contain the same fluid.
The fluids are static and density is constant.
The measured height is vertical.

1

Start from a pressure walk

The manometer-fluid column and process-fluid column act in opposite directions.

P_{1} - P_{2} = ρ_{m} g h - ρ g h

2

Factor common terms

The pressure difference depends on the density difference between the two fluids.

P_{1} - P_{2} = (ρ_{m} - ρ) g h

Result

P_{1} - P_{2} = (ρ_{m} - ρ) g h

Source: Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013; Engineering LibreTexts, 4.3.2.3: Magnified Pressure Measurement, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $h = \frac{P _{1} - P _{2}}{( ρ _{m} - ρ ) g}$

Height Difference

\frac{P _{1} - P _{2}}{( ρ _{m} - ρ ) g} = h

Divide both sides by the product of the density difference and gravity to isolate the height difference.

Difficulty: 2/5

Solve for $ρ = ρ_{m} - \frac{P _{1} - P _{2}}{g h}$

Process Fluid Density

ρ = ρ_{m} - \frac{P _{1} - P _{2}}{g h}

Isolate the density term by dividing by gh, then rearranging the subtraction.

Difficulty: 3/5

Solve for $ρ_{m} = ρ + \frac{P _{1} - P _{2}}{g h}$

Manometer Fluid Density

ρ_{m} = ρ + \frac{P _{1} - P _{2}}{g h}

Isolate the density term by dividing by gh, then adding the process density to both sides.

Difficulty: 3/5

Solve for $g = \frac{P _{1} - P _{2}}{( ρ _{m} - ρ ) h}$

Gravitational Acceleration

g = \frac{P _{1} - P _{2}}{( ρ _{m} - ρ ) h}

Divide both sides by the product of density difference and height.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph shows a straight line with a positive slope, illustrating how pressure difference changes with height difference. For a student, this means that if you increase the height difference in a differential manometer, the pressure difference will also increase proportionally, provided the fluid densities and gravity remain the same. The most important feature is the linear relationship, directly reflecting the formula's structure. This visualizes how a larger height difference directly leads to a larger pressure difference.

Graph type: linear

Why it behaves this way

Intuition

The manometer compares two columns. The pressure difference is created by the extra weight of the manometer liquid relative to the process fluid.

P_{1} - P_{2}

Pressure difference

How much higher one tap pressure is than the other.

ρ_{m} - ρ

Density difference

The extra heaviness that makes the manometer respond.

h

Column height difference

The visible vertical offset in the manometer liquid.

Free study cues

Insight

Canonical usage

This equation is used to calculate the pressure difference between two points in a fluid by measuring the difference in height of a manometer fluid column.

Common confusion

Students may confuse the density of the manometer fluid with the density of the process fluid, or use inconsistent units for height and density.

Dimension note

This equation involves physical quantities with units; the result is not dimensionless.

Unit systems

$P_{1} - P_{2}$ Pa · Pressure difference is the primary output and is typically expressed in Pascals (Pa).

$ρ_{m}$ kg/m^3 · Density of the manometer fluid.

$ρ$ kg/m^3 · Density of the process fluid.

$g$ m/s^2 · Standard gravitational acceleration is approximately 9.80665 m/s^2.

$h$ m · The difference in height of the manometer fluid columns.

One free problem

Practice Problem

A differential manometer uses mercury with density 13600 kg/ $m^{3}$ to measure a water line with density 1000 kg/ $m^{3}$ . If h = 0.080 m and g = 9.81 m/ $s^{2}$ , what is P1 - P2?

Manometer Fluid Density13600 kg/m^3

Process Fluid Density1000 kg/m^3

Gravitational Acceleration9.81 m/s^2

Height Difference0.08 m

Solve for: pressureDifference

Hint: Use the density difference.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When measuring pressure drop across a pipe fitting using a mercury U-tube manometer, Differential manometer is used to calculate Pressure Difference from Manometer Fluid Density, Process Fluid Density, and Gravitational Acceleration. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Study smarter

Tips

Use the density difference, not just the manometer-fluid density.
Use vertical height difference.
The sign depends on which pressure is defined as P1.

Avoid these traps

Common Mistakes

Using $r h o_{m}$ alone when the process-fluid density is not negligible.
Swapping P1 and P2 without changing the sign convention.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The simplified differential manometer equation comes from the general pressure balance when both connection legs contain the same process fluid.

Use this simplified form for a U-tube differential manometer connected between two points in the same fluid.

The equation turns a visible column-height difference into a pressure difference across a pipe, filter, or restriction.

Using rho_m alone when the process-fluid density is not negligible. Swapping P1 and P2 without changing the sign convention.

When measuring pressure drop across a pipe fitting using a mercury U-tube manometer, Differential manometer is used to calculate Pressure Difference from Manometer Fluid Density, Process Fluid Density, and Gravitational Acceleration. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Use the density difference, not just the manometer-fluid density. Use vertical height difference. The sign depends on which pressure is defined as P1.

References

Sources

Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013
Engineering LibreTexts, 4.3.2.3: Magnified Pressure Measurement, accessed 2026-04-09
NIST CODATA
IUPAC Gold Book
Wikipedia: Differential manometer
Textbook: Fluid Mechanics by Frank M. White
NIST Chemistry WebBook
University Physics with Modern Physics by Hugh D. Young and Roger A. Freedman

Differential manometer