Bulk Modulus and Compressibility

E_{v} = - v \frac{d p}{d v} = ρ \frac{d p}{d ρ}, c = \frac{E _{v}}{ρ}

Ev of water is about 2.2 GPa — liquid is nearly incompressible for Civil FE except water hammer (c=√(Ev/ρ) reduced by pipe wall). Compressibility β=1/Ev.

$E_{v}$ (Pa): Bulk modulus
$c$: Celerity of sound / surge

Step-by-step solved example

Water Ev=2.2 GPa, ρ=1000 kg/m³. Acoustic c (rigid pipe)?

A pressure pulse (water hammer) travels at c = √(Ev/ρ).

1. c
√(2.2e9/1000)=√2.2e6=1483 m/s.
$c \approx 1480 m/s$
2. Comment
Real pipes are slower (Joukowsky still uses this c).

Answer: c ≈ 1480 m/s

10 practice questions

0/10 correct

1.β compressibility is

2.Δp=Ev Δρ/ρ so 0.1% density change in water needs

3.Joukowsky Δp ≈ ρ c ΔV for

4.Ideal gas isothermal Ev equals

5.c=√(kRT) is

6.If the pipe wall is elastic, c

7.Units of Ev are

8.Incompressible assumption on FE Civil is OK when

9.ρ=1000, c=1000 m/s, ΔV=2 m/s: Δp ≈

10.Bulk modulus is the volumetric analog of