Summary

This article provides the formulae for the calculation of speed of sound in fluids and fluid filled circular pipes. The speed of sounds is important in piping systems for the calculation of choked flow for gases and pressure transient analysis of liquid filled systems.

Definitions


$a$	:	Speed of sound in fluid
$D$	:	Internal diameter of pipe
$E$	:	Bulk modulus of pipe
$e$	:	Pipe wall thickness
$K$	:	Bulk modulus of fluid
$L$	:	Length of the pipe
$t$	:	Travel time for pressure wave
$\psi$	:	Pipe support factor
$\nu$	:	Poisson’s ratio of pipe material

Introduction

The speed of sound (also known as wave celerity or phase speed) is the speed at which a pressure wave travels in a given medium. The speed of sound is a function of a fluid’s density $\rho$ and bulk modulus $K$ .

Determination of the speed of sound allows the prediction of how long it will take a wave (such as a sudden pressure change) to propagate through a system and is therefore a necessary parameter in transient analysis. For example, if a valve closes at the end of a long pipeline the time it takes to observe an increase in pressure at the start of the pipeline may be calculated by dividing the pipe length by the speed of sound.

Speed of Sound for Fluids

For the general case the speed of sound in liquids and gases can be calculated using the Newton-Laplace equation:

$a = \sqrt{\frac{K}{\rho}}$

The bulk modulus of a range of fluids may be found in this article.

Speed of Sound for Fluids in Pipes

Rigid Pipes

For an ideal, perfectly rigid pipe, the general speed of sound can be calculated using the Newton-Laplace equation as shown above.

Circular Elastic Pipe

For a pipe with elastic walls the speed of sound in the fluid is impacted by the elasticity of the pipe wall and the pipe supports.

$a = \sqrt{\frac{1}{\rho \left( \frac{1}{K} + \frac{D \psi}{E e} \right)}}$

Calculation of the pipe support factor varies based on how the pipe is supported. The following section includes a list of correlations to determine the pipe support factor for several cases.

Pipe Support Factors

The pipe support factor is a function of the Possion’s ratio of the pipe material ( $\nu$ ), the pipe diameter ( $D$ ) and the pipe wall thickness ( $e$ ). For the calculation of the pipe support factor a pipe is considered thin walled if $D / e > 10$ .

When the pipe material has a large bulk modulus, as is the case for steel piping, or frequent expansion joints have been installed along the line, the pipe support factor can typically be taken as unity without significant error.

Case	Equation
Thin-walled pipe free to expand throughout	$\psi = 1$
Thin-walled pipe anchored at upstream end only	$\psi = 1 - 0.5 \nu$
Thin-walled pipe anchored throughout	$\psi = 1 - \nu^{2}$
Thick-walled pipe free to expand throughout	$\psi = \dfrac{2e}{D}(1+\nu) + \dfrac{D}{D+e}$
Thick-walled pipe anchored at upstream end only	$\psi = \dfrac{2e}{D}(1+\nu) + \dfrac{D}{D+e}\left(1 - \dfrac{\nu}{2}\right)$
Thick-walled pipe anchored throughout	$\psi = \dfrac{2e}{D}(1+\nu) + \dfrac{D}{D+e}(1 - \nu^{2})$
Circular tunnel	$\psi = \dfrac{2e}{D}(1+\nu)$

Pressure Wave Communication Time

The time it takes for a pressure wave to travel from its origin (i.e. sudden valve closure) to a source (pressure transmitter) is known as the communication time. Communication time can be calculated from the speed of sound using the following equation.

$\Delta t = \frac{2L}{a}$

Communication time is an important parameter in transient analysis as any event that occurs within a time frame shorter than the communication time is equivalent to an instantaneous event.

Speed of Sound in Fluids and Fluid in Pipes