---
title: "Water mass for pressurised vessel"
subject: "Fluid Mechanics"
chapter: 1
key: "semester_3/fluid_mechanics:pressure-vessel-bulk-modulus"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/pressure-vessel-bulk-modulus
site: BCT Engineering
---

# Water mass for pressurised vessel

Fluid Mechanics · Chapter 1: Fluid and its physical properties

- Asked in: 2073 Shrawan, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/pressure-vessel-bulk-modulus
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-1

## Question

A pressure vessel has an internal volume of 0.5 m³ at atmospheric pressure. It is desired to test the vessel at 3000 bar by pumping water into it. The estimated variation in the change of the empty volume of the container due to pressurization to 3000 bar is 0.6 percent. Calculate the mass of water to be pumped into the vessel to attain the desired pressure level given the bulk modulus of water as 2000 MPa.

## Short answer

The vessel expands 0.6% to 0.503 m³ at 3000 bar. Water compresses by 15% (ΔV/V = 300/2000), so atmospheric volume needed is 0.503/0.85 = 0.5918 m³. Mass = 1000 × 0.5918 ≈ 592 kg.

## Answer

**Data:** $V_0=0.5\ \text{m}^3$, test pressure $\Delta p=3000\ \text{bar}=300\ \text{MPa}$, $K=2000$ MPa, the empty vessel expands by 0.6 %.

**Step 1: volume of the vessel at 3000 bar** (expanded by 0.6 %):

$$
V_v=0.5\times1.006=0.503\ \text{m}^3
$$

**Step 2: compression of water.** From $K=-\dfrac{\Delta p}{\Delta V/V}$:

$$
\frac{\Delta V}{V}=\frac{\Delta p}{K}=\frac{300}{2000}=0.15
$$

So the water occupies 85 % of its original (atmospheric) volume at 3000 bar. The water must fill the vessel (0.503 m³) at that pressure:

$$
V_w(1-0.15)=0.503\ \Rightarrow\ V_w=\frac{0.503}{0.85}=0.5918\ \text{m}^3
$$

**Step 3: mass of water** (density of water at atmospheric pressure 1000 kg/m³):

$$
m=\rho V_w=1000\times0.5918=591.8\ \text{kg}
$$

**Answer: about 592 kg of water must be pumped in** (about 92 kg more than the 500 kg that would fill the vessel at atmospheric pressure).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With K = 2000 MPa and a pressure rise of 300 MPa, what is the fractional volume decrease of water?
   A. 0.015
   B. 0.15
   C. 1.5
   D. 6.67
   Answer: B. Delta V/V = Delta p/K = 300/2000 = 0.15. (Key point: Compression of water)

2. Which equation gives the original water volume V_w needed?
   A. V_w x 1.15 = 0.503
   B. V_w = 0.503 x 0.85
   C. V_w (1 - 0.15) = 0.503
   D. V_w = 0.5 x 0.85
   Answer: C. After compression the water occupies 0.85 V_w, which must fill 0.503 m^3, so V_w = 0.5918 m^3. (Key point: Water volume to pump)

3. What mass of water must be pumped in?
   A. About 592 kg
   B. About 503 kg
   C. About 500 kg
   D. About 428 kg
   Answer: A. m = 1000 x 0.5918 = 591.8 kg using the atmospheric density. (Key point: Mass of water)

## Related in Fluid Mechanics

- [Time to discharge pressurised tank](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Apressurised-tank-emptying-orifice) — Fluid Mechanics, chapter 6
- [Pressure change from displacement thickness in duct](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aduct-displacement-thickness-pressure) — Fluid Mechanics, chapter 8
- [Rotating U-tube pressure and water lost](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Arotating-u-tube-suction-leak) — Fluid Mechanics, chapter 2
- [Absolute and gauge pressure, mercury manometer](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aabsolute-gauge-pressure-vapour-mercury) — Fluid Mechanics, chapter 2
- [Accelerated oil tank, spilled volume](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aaccelerated-cube-tank-spilling) — Fluid Mechanics, chapter 3
- [Oil-water manometer with gas cylinder](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aoil-water-manometer-gas-cylinder) — Fluid Mechanics, chapter 2

## Related in other subjects

- [Cylindrical vessel 3 m long: change in volume](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acyl-vessel-3m-volume-change) — Strength of Materials, chapter 5
- [Spherical vessel, volume increase 300 cm3, find pressure](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aspherical-vessel-300cm3) — Strength of Materials, chapter 5, asked 2×
- [Rigid container heated from 1 bar to 1.5 bar](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Arigid-container-1bar-to-1-5bar-quality-0-5) — Fundamental of Thermodynamics & Heat Transfer, chapter 3, asked 2×
- [Cylindrical shell: volume increase, 6 N/mm2 pressure](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acyl-shell-4m-300mm-6nmm2-volume) — Strength of Materials, chapter 5