Acta Polytechnica


doi:10.14311/AP.2015.55.0422
Acta Polytechnica 55(6):422–426, 2015 © Czech Technical University in Prague, 2015

available online at http://ojs.cvut.cz/ojs/index.php/ap

ON THE SPEED OF SOUND IN STEAM

Pavel Šafaříka, ∗, Adam Novýb, David Jíchaa, b, Miroslav Hajšmanb

a Czech Technical University in Prague, Faculty of Mechanical Engineering, Technická 4, 166 07 Prague 6, Czech
Republic

b Doosan Škoda Power, Tylova 1/57, 301 28 Plzeň, Czech Republic
∗ corresponding author: pavel.safarik@fs.cvut.cz

Abstract. A study of the speed of sound in a pure water substance is presented here. The IAPWS
data on the state of water and steam are applied only for investigating the speed of sound for a
one-phase medium. A special numerical model for investigating the parameters of shock waves in steam
is presented here and is applied for investigating extremely weak waves to obtain velocities representing
the speed of sound in both one-phase and two-phase steam. Problems with the speed of sound in
two-phase steam are discussed, and three types of speed of sound are derived for the metastable region
of wet steam.

Keywords: speed of sound; steam; IAPWS-IF97; shockwave.

1. Introduction
The speed of sound is a physical quantity closely con-
nected with the compressibility of a medium. Physi-
cally, the speed of sound expresses the speed of the
advance of undulation in a given medium. Generally,
the speed of sound a in a continuum is defined as the
square root of an infinitesimal pressure disturbance
∂p related to an infinitesimal change in density ∂% in
an isentropic process

a =

√(∂p
∂%

)
s
. (1)

Thee fundamental definition of the speed of sound
is expressed by (1). There is no problem in evaluating
the speed of sound when the state equation in the
form f(p,%,s) = 0 is known. The well-known relation
of the speed of sound can be derived for an ideal gas
as

a =
√
κp/% =

√
κrT. (2)

From (2), it follows that the speed of sound in an
ideal gas depends on temperature only, because κ =
const. is the ratio of the heat capacities (the Poisson
constant) and r is the specific gas constant

r = R/M, (3)

R = 8314.41 J kmol−11 K−1 is the universal gas con-
stant, and M is the molar mass of the gas.

Some considerations on the speed of sound in steam
will be presented in this paper. In their previous pub-
lications [1, 2], the authors pointed out problems with
the propagation of waves in steam. One point is that
the speed of sound in steam depends on state param-
eters in a more complicated manner than in (2). This
is obvious, because the equation of state for steam
according to the data of the International Association

for Properties of Water and Steam (IAPWS) is com-
plex. IAPWS has released two formulations for water
and steam: the first is the Formulation for General
and Scientific Use IAPWS-95 [3], and the second is
the Industrial Formulation IAPWS-IF97 [4].
It should be mentioned here that the data on the

speed of sound in steam are available only for a one-
phase medium. The available tools are tables or calcu-
lators. The data on the speed of sound in wet steam
have not yet been integrated. Papers [5, 6] provide
much stimulation for further studies of the speed of
sound in wet steam, because they deal with the prop-
agation of waves in wet steam. However, no measured
data have in fact been published.

2. Speed of sound in steam
The formulation of IAPWS-95 is a fundamental equa-
tion for specific Helmholtz free energy f(%,T). Its
dimensionless form is separated into two parts

f(%,T)
rT

= φ(δ,τ) = φo(δ,τ) + φr(δ,τ), (4)

where δ = %/%c and τ = Tc/T. For water substance
reference constants, IAPWS defined in [3] the crit-
ical temperature Tc = 647.096 K, the critical den-
sity %c = 322 kg m−3, and the specific gas constant
r = 461.51805 J kg−1 K−1. Functions φo(δ,τ) and
φr(δ,τ) are defined by IAPWS [2]. The speed of
sound is then calculated from

a(δ,τ) =
[
rt

(
1 + 2δ

∂φr

∂δ
+ δ2

∂2φr

∂δ2

−
(
1 + δ∂φ

r

∂δ
− δτ ∂

2φr

∂δ ∂τ

)2
τ2
(
∂2φo

∂τ2
+ ∂

2φr

∂τ2

) )
]1/2

. (5)

The formulation of IAPWS-IF97 is divided into 5
regions, where different fundamental equations are

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vol. 55 no. 6/2015 On the Speed of Sound in Steam

Figure 1. Values for the speed of sound [m s−1] in
water and in steam in a p–t diagram.

defined: Gibbs free energy g(p,T) and Helmholtz free
energy f(%,T). The fundamental equation for Gibbs
free energy is expressed in dimensionless form as

g(p,T)
rT

= γ(π,τ), (6)

where π is dimensionless reduced pressure, and τ is
dimensionless reduced temperature. The function
γ(π,τ) is defined by IAPWS [4], and then the speed
of sound is solved from

a(π,τ) =

√√√√ rT(∂γ∂π)2τ2 ∂2γ∂τ2(
∂γ
∂π
− τ ∂

2γ
∂π ∂τ

)2 − ∂2γ
∂π2

τ2 ∂
2γ
∂τ2

. (7)

Values for the speed of sound were evaluated according
to (7), and were presented in [1] in a p–t (pressure-
temperature) phase diagram. The values are shown
in Fig. 1. It is evident that ideal gas theory cannot
be applied for water and steam.

3. The speed of sound in steam
solved by means of the model
for solving the thermodynamic
parameters of steam
downstream from a normal
shock wave

An equilibrium model of a shock wave in steam was
formulated in [1]. The theoretical approach for calcu-
lating steam parameters is based on balance equations
for steam passing the infinitesimally thin control vol-
ume on a normal shock wave (Fig. 2). The three mod-
ified balance equations are: balance of mass, balance
of momentum, and balance of energy (under the as-
sumption of constant total enthalpy h01 = h02 = h0):

Figure 2. Scheme of a shock wave, control volume,
and parameters on a normal shock wave in steam.

• balance of mass

ṁ

A
= %1v1 = %2v2; (8)

• balance of momentum p1 −p2 = ṁA (v2 −v1), hence

p1 + %1v21 = p2 + %2v
2
2 ; (9)

• balance of energy

h1 +
v21
2

= h2 +
v22
2

= h01 = h02 = h0; (10)

• equation of state

1/% = fv,ph(p,h). (11)

Index 1 is upstream from the shock wave, index 2 is
downstream from the shock wave, 0 indicates a total
value.

All thermodynamic parameters upstream from the
normal shock wave are given. In paper [1], the calcula-
tion procedure for a given pressure downstream from
the shock wave p2 (when p2 > p1) is derived. The iter-
ative procedure is based on the balance equations and
the equation of state of steam according to IAPWS-
IF97, and all thermodynamic parameters downstream
from the normal shock wave are calculated. The wet
steam model assumes isobaric separation of the two-
phase medium, for further details, see [1].
Relations for velocities as functions of the calcu-

lated thermodynamic parameters can be derived from
the balance equations. The velocity v1 of steam up-
stream from the normal shock wave can be calculated
according to the relation

v1 =
p2−p1
%1√

2p2−p1
%1
− 2(h2 −h1)

. (12)

The velocity v2 downstream from the normal shock
wave is also derived from the balance equations, and
can be calculated according to the relation

v2 = v1 −
p2 −p1
%1v1

. (13)

The model for calculating the shock wave parameters
was applied successfully for superheated, saturated
and wet steam for various ratios of pressures p2/p1.
A special case is for p2 = p1; then v1 = v2 = a, (a is
the speed of sound).

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P. Šafařík, A. Nový, D. Jícha, M. Hajšman Acta Polytechnica

Figure 3. The h–s diagram of water and steam with curves of the constant speed of sound in steam a [m s−1]. The
wet steam region blue lines, are from definition (1) and the blue cross values are calculated by means of the model
for normal shock waves. Superheated steam a values according to IAPWS-IF97 are depicted by green curves.

3.1. Results of the solution
for the speed of sound

The numerical procedure for the model for calculat-
ing the thermodynamic parameters of steam down-
stream from a normal shock wave was performed for
p2/p1 � 1. The values obtained for the speed of sound
proved to be very close to the IAPWS data, accord-
ing to (7) for superheated and saturated steam. We
developed calculation tool in MatLab, which proved
to be well performing. The results for wet steam are
very close (lower than 2 % difference) to the values for
the speed of sound in wet steam, derived according
to (1), see Fig. 3, where a detail of the h–s (enthalpy–
entropy) diagram for water and steam is depicted.
The assumption of an infinitesimally weak normal
shock wave defines the equilibrium conditions for wet
steam, so the speed of the sound values obtained here
corresponds to the speed of sound for the thermody-
namic equilibrium state of wet steam. A numerical
model for calculating the speed of sound in steam is
developed and verified.
Figure 3 shows some special effect of the speed of

sound on the steam saturation line – there is consid-
erable discontinuity of the values. This effect can be
explained by a discontinuity of the first derivative of
lines of constant temperature on the steam satura-
tion line in the h–s (enthalpy–entropy) diagrams and
in the p–v (pressure–specific volume) diagrams, and
perhaps by a discontinuity of the first derivative of
the lines of constant pressure on the saturation line
in the T–s (temperature–entropy) diagram for steam.
The state parameters — namely pressure p, density %
and entropy s — defined in the two-phase region by
means of the Maxwell rule as equilibrium parameters.
From the thermodynamic point of view, wet steam in
the equilibrium state is therefore considered to be a
continuum, which contains both phases of water (sat-
urated water liquid and saturated steam) according
to the dryness value. The speed of sound obtained
from this numerical model for wet steam should be
referred to as the equilibrium speed of sound.

4. The speed of sound in wet
steam in the metastable region

4.1. Equilibrium speed of sound
The equilibrium speed of sound can be evaluated for
the whole wet steam region, see Fig. 3. Low equi-
librium speed of sound values near the saturation
line of water are not acceptable. This is shown in
Fig. 4, where the speed of sound values are depicted
as dependencies of dryness x for constant pressures p.

4.2. Frozen speed of sound
The assumption is often made that isobars and
isotherms are identical in the wet steam region when
calculating the speed of sound in wet steam. Accord-
ing to (2), the value of the speed of sound is therefore
calculated using the relation

aF =
√
κrT ′′. (14)

where T ′′ is temperature of saturated steam. The lines
for the constant speed of sound are therefore identical
to the isobars in wet steam. The speed of sound
calculated using this assumption should be referred
to as the frozen speed of sound. This approach can be
applied for approximate calculations in the region of
temperatures up to 100 °C and dryness of wet steam
from 0.98 to 1.00. However, it is more convenient to
determine the frozen speed of sound from the IAPWS
data for saturated steam.

4.3. Metastable speed of sound
A further approach is based on a continuum model,
so that the dependence of the speed of sound on
the specific enthalpy is extended from the region of
superheated steam into the region of wet steam. If
superheated steam is considered as an ideal gas, the
dependence of speed of sound on specific enthalpy is
parabolic. It can be derived from (2):

aMC =
√

(κ− 1)hid, (15)

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vol. 55 no. 6/2015 On the Speed of Sound in Steam

Figure 4. Dependence of values for the equilibrium speed of sound a on the dryness of wet steam at constant
pressures p.

Pressure p Specific enthalpy h Dryness x
[MPa] [kJ kg−1] [kgsat kgwet−1]
0.001 2421.89 0.9629
0.010 2473.12 0.9534
0.100 2542.29 0.9411
1.000 2645.18 0.9350
2.000 2658.32 0.9264
5.000 2660.51 0.9186
10.000 2555.87 0.8699

Table 1. Parameters of wet steam where the
metastable and equilibrium speeds of sound are
equal.

where κ is the ratio of specific heat capacities and hid
is the specific enthalpy of an ideal gas defined as a zero
value at zero total temperature. The IAPWS-IF97
data are depicted in Fig. 5 in the diagram showing
speed of sound vs. specific enthalpy for constant
pressures. In the region of superheated steam the
dependencies are similar curves, but the equilibrium
speed of sound has a discontinuity on the saturation
line. It should be mentioned here that the specific
enthalpy of steam is applied according to the IAPWS
definition – in the triple point, the enthalpy of steam is
ht.p. = 2500.9 kJ kg−1. In the new model, the curves
are prolonged into the region of wet steam. The speed
of sound defined by these prolonged curves should be
referred to as the metastable speed of sound. The
region from the saturation line of steam to the inter-
section of the prolonged curves with the dependencies
of the equilibrium speed of sound determines the re-
gion of metastable speed of sound. We estimated the
numerical uncertainty of the calculation to be lower
than 5 % and can be further improved. The conditions
for equality values of the metastable and equilibrium

speeds of sound are determined and introduced in
Table 1.

It is remarkable that the dryness values in Table 1
are very close to the Wilson line for condensation
of steam into droplets. The speed of sound in the
metastable region can acquire any value between the
metastable speed of sound and the equilibrium speed
of sound. The values for the frozen speed of sound
are rather overvalued. For lower values of the dry-
ness of wet steam given in Table 1, continuum models
lose validity, and it is necessary to investigate the
propagation of waves in a heterogeneous (two-phase)
environment. Examples can be found in [5, 6]. The
metastable-vapor region is defined by IAPWS [4, 7]
from the saturation line to the 95 % equilibrium dry-
ness line for pressures from the triple-point pressure
up to 10 MPa.

5. Conclusions
A numerical model for calculating the speed of sound
in steam based on balances of mass, momentum and
energy and an equation of state for steam based on
IAPWS-IF97 has been developed and verified. Our
results are in very good accordance with the IAPWS
data in the region of superheated steam, where the
relative error is ranging from 0.3 % up to the maxi-
mum of 1.5 % in the critical point surrounding. Proper
method for determining the uncertainty of the calcu-
lated values of the speed of sound in wet steam was
not yet determined. A detailed study of the speed of
sound in wet steam has attempted to determine the
metastable region and to define the speed of sound
in this region. It should be pointed out here that no
measured data are available for the speed of sound in
wet steam, and there are also only limited theoretical
resources.

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P. Šafařík, A. Nový, D. Jícha, M. Hajšman Acta Polytechnica

Figure 5. Dependence of speed of sound on specific enthalpy for constant pressures.

List of symbols
p pressure [Pa]
a speed of sound [m s−1]
% density [kg m−3]
T temperature [K]
t temperature [°C]
κ ratio of heat capacities (Poisson constant) [1]
r specific gas constant [J kg−1 K−1]
v velocity [m s−1]
h specific enthalpy [kJ kg−1]
s specific entropy [kJ kg−1 K−1]
x dryness [kgsat kg−1wet]
A area [m2]
ṁ mass flux [kg s−1]

Acknowledgements
Support from the Technology Agency of the Czech Repub-
lic under project TE01020036 is gratefully acknowledged.
Doosan Škoda Power also provided crucial support for this
research.

References
[1] ŠAFAŘÍK, P., NOVÝ, A., HAJŠMAN, M., JÍCHA, D.
On a model for solution of shock wave parameters in wet
steam, Paper No.PWS-011. In: 16th International
Conference on Properties of Water and Steam,
Proceedings (electronic form), London: IMechE, 2013.

[2] NOVÝ, A., JÍCHA, D., ŠAFAŘÍK, P., HAJŠMAN, M.
On parameters of shock waves in saturated steam. pp.
43-46. In: Topical Problems of Fluid Mechanics 2013,
Proceedings, Prague, 2013.

[3] IAPWS Revised release on the IAPWS formulation
1995 for the thermodynamic properties of ordinary
water substance for general and scientific use, IAPWS
Release, 2014.

[4] IAPWS Revised release on the IAPWS industrial
formulation 1997 for the thermodynamic properties of
water and steam, IAPWS Release, 2007.

[5] PETR, V. Wave propagation in wet steam, Journal of
Mechanical Engineering Science, Vol.218, Part C,
pp.871-882, 2004. doi:10.1243/0954406041474237

[6] YOUNG, J. B., GUHA, A. Normal shock-wave
structure in two-phase vapor-droplet flows, Journal of
Fluid Mechanics, Vol.228, pp.243-274, 1991.
doi:10.1017/S0022112091002690

[7] WAGNER, W., COOPER, J.R., DITTMANN, A.,
KIJIMA, J., KRETZSCHMAR, H.-J., KRUSE, A.,
MAREŠ, R., OGUCHI, K., SATO, H., ST’OCKER, I.,
ŠIFNER, O., TAKAISHI, Y., TANISHITA, I.,
TRÜBENBACH, J., WILLKOMMEN. Th. The IAPWS
industrial formulation 1997 for the thermodynamic
properties of water and steam, Journal of Engineering
for Gas Turbines and Power, Transaction of the ASME,
Vol.122, pp.150-182, 2000, doi:10.1115/1.483186

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	Acta Polytechnica 55(6):422–426, 2015
	1 Introduction
	2 Speed of sound in steam
	3 The speed of sound in steam solved by means of the model for solving the thermodynamic parameters of steam downstream from a normal shock wave
	3.1 Results of the solution for the speed of sound

	4 The speed of sound in wet steam in the metastable region
	4.1 Equilibrium speed of sound
	4.2 Frozen speed of sound
	4.3 Metastable speed of sound

	5 Conclusions
	List of symbols
	Acknowledgements
	References