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7. Speed of sound as a thermodynamic property of fluids
7. SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS Daniel G. Friend Physical and Chemical Properties Division Chemical Science and Technology Laboratory National Institute of Standards and Technology Boulder, Colorado
Abstract In this Chapter, we review the principles of sound propagation in fluid systems. From a study of the hydrodynamic equations, sound propagation is shown to be a wave phenomenon. The speed of sound then can be derived at any state point from a knowledge of the thermodynamic surface of the fluid of interest. Several model equations of state are reviewed, and it is shown how the speed of sound can be obtained for a variety of systems. We then focus on several fluids of particular interest, and show the behavior of the sound speed over a wide range of the temperature and pressure variables. Tabulated values of the speed of sound are given for argon, nitrogen, water, and air based on the current standard reference thermodynamic surfaces.
7.1 Introduction In this Chapter, we discuss the propagation of sound in fluids and provide information about the thermodynamic speed of sound over substantial ranges of the state variables for a variety of fluids. In the context of this chapter, we consider sound to arise from a small periodic and isentropic (constant entropy) perturbation of the local equilibrium in a fluid, which, as we shall see, gives rise to a standard wave equation. The systems under consideration include both pure fluids and mixtures in the liquid, vapor, and supercritical states. Thus the range in temperature is from the melting line to very high temperatures (a dissociation limit), and the range in pressure is from very low values (below which the continuum approximation would not be valid) to the solidification locus (at least in principle). The major theme of the discussion stems from the relation 2-1
W = <
\dpdTj
M \
dp \
dp^
(7.1)
- p-
ar2
p
1/2
-J
237 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Vol. 39 ISBN 0-12-475986-6
Copyright © 2001 by Academic Press All rights of reproduction in any form reserved. ISSN 1079-4042/01 $35.00
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SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
which expresses the speed of sound, w, in terms of derivatives of the molar Helmholtz free energy A and the state point defined by the temperature T and the molar density p. In Eq. 7.1, M is the molar mass of the substance in kilograms per mole. Although Eq. 7.1 may not be the simplest expression for the speed of sound, it serves to emphasize that this thermodynamic quantity is easily calculable if the Helmholtz free energy of a fluid system is known. This expression is also appropriate for fluid mixtures, in which case the partial derivatives are to be taken at constant composition. In the next section, we outline how Eq. 7.1 can be obtained. Although this thermodynamic relationship is straightforward and exact, a few caveats must be given regarding its use. First "thermodynamic" sound model requires isentropic and linear approximations, and thus there are restrictions on the amplitude and frequency of the propagated disturbance. These issues are seldom significant when considering sound in most fluid media under most conditions. The second caveat is more essential: there are only a few fluid systems for which the Helmholtz free energy surface is sufficiently well known to allow sound speeds to be accurately calculated. These surfaces are generally empirical correlations of experimental data, and only when accurate sound speeds are included in the primary data used to determine the thermodynamic surface are the resultant calculated sound speeds obtained with small uncertainty. Experimental determinations of sound speeds, using state-of-theart techniques as discussed in other chapters, usually produce more accurate values for this quantity. However, more theoretical or predictive models for the Helmholtz free energy surface may be used to obtain useful approximations for the speed of sound for systems that have not been thoroughly characterized through experimental measurements. Once we have completed a derivation of Eq. 7.1 in the next section, we will explore some of the limits of this approach, and discuss the application to various model systems and to real fluids. Table 7.1 included at the end of this chapter provides reliable speed of sound values for a variety of important fluid systems.
7.2 Wave Equation in a Fluid From a standard understanding of thermodynamic conventions, it may be difficult to see how sound propagation can be considered a thermodynamic property. Despite a straightforward parsing of the name "thermodynamics," typical thermodynamic studies deal with unperturbed systems. Other responses to perturbing stresses, such as flow associated with shear stresses, heat flux related to thermal gradients, and diffusive flows caused by composition variations, are not considered thermodynamic in origin, and the separate study of steady-state kinetic theory may be required to obtain viscosity, thermal
WAVE EQUATION IN A FLUID
239
conductivity, and diffusion coefficients. The measurement of thermodynamic properties, as with most other physical quantities, may require a perturbing probe (i.e., a temperature change to determine heat capacities), but, in general, thermodynamic processes are equilibrium phenomena. As we will see, it is the linear, reversible, and small amplitude nature of the wave process that serves to connect sound propagation with fluid thermodynamics. 7.2.1 Hydrodynamic Equations When fluids are considered in the continuum approximation, three conservation laws can be used to determine the response to perturbing influences in an otherwise equilibrated system. These hydrodynamic equations enforce the conservation of mass (or the number of molecules of each species present), the conservation of momentum, and the conservation of energy. The continuity equation, ^
+ V . (pu) = 0
(7.2)
ensures that the molar density (at a given fixed infinitesimal volume in space) can change only with a flow of molecules into or out of the volume; here, the partial derivative is with respect to time t, and u represents the macroscopic velocity of fluid flow at the given point. We use the Euler convention in which the frame of reference is fixed in space, so that properties such as p are functions of position (r) and time, and they represent microscopic averages within a volume element centered on r in this fixed reference frame. The Navier-Stokes equation (7.3) f I - + u • v ) u = - - T ^ V(P - r;^ V • u) + - ^ V^u \dt J m Mp Mp is equivalent to Newton's second law and balances changes in momentum with forces acting on the fluid element. The operator ( ^ + u • V) is the material derivative or streaming derivative, and the left side of the equation describes changes in momentum due directly to fluid flow. The forces that act to change the momentum include the external force F, here divided by the mass m contained in the infinitesimal volume element under consideration, the force on the element due to gradients of pressure P, and forces associated with the viscous nature of any fluid: here, rjs is the bulk viscosity and rj is the usual shear viscosity. The pressure tensor has been decomposed, and a linear relationship between the stress and shear has been assumed [1]. Finally, the conservation of energy can be written in the form / d
\
R
X
^
240
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
where T is the absolute temperature, R is the molar gas constant, C^ is the molar isochoric heat capacity, and X is the coefficient of thermal conductivity in a linear, Fourier's law, relationship between thermal gradients and heat flux. Equations 7.2 to 7.4 are often simplified in the basic development of a wave equation formalism. In particular, we first consider an ideal fluid with no viscous effects or thermal conduction. In the absence of external forces, the Navier-Stokes relationship given by Eq. 7.3 reduces for an ideal fluid to the Euler equation.
Small disturbances of an equilibrium state can generate sound, and in the derivation of the wave equation, second-order effects in the derivatives can be ignored. Because the equilibrium state corresponds to zero macroscopic velocity, any quadratic terms in u will also be dropped. These approximations are almost always quite satisfactory in typical acoustic situations, and we will provide some quantitative discussion below. To lowest order, Eq. 7.5 can then be linearized and written as VP + M p , ^ = 0
(7.6)
where we have now introduced the notation XQ to denote the time-invariant and spatially invariant equilibrium value of a quantity. (In this notation, a physical quantity can be represented by x = x^ + X A , where XA is then the "acoustic" variable; i.e.. PA is the acoustic pressure, or the deviation of the pressure from its equilibrium value caused by a perturbation.) To the same order, Eq. 7.2 is written as ^
+ p„V • u = 0
(7.7)
01
Taking the time derivative of Eq. 7.7 and the divergence of Eq. 7.6 and subtracting the results gives M ^ - V 2 p = 0
(7.8)
Equation 7.8 forms the basis of the wave equation when we recall that the pressure and density can be related through the thermodynamic equation of state or Helmholtz energy surface. For a pure fluid, the pressure can be considered to be a function of the density and a second intensive thermodynamic variable (and for mixtures, the composition must also be given); without specifying the second variable, we write P = P(p,a), so that derivatives of the
WAVE EQUATION IN A FLUID
241
pressure can be written in terms of derivatives of the density and the parameter ot. Again we drop some higher-order terms and write Eq. 7.8 as M— 3/2
vV - V.
(7.9)
dp
If we can identify an appropriate variable a which remains constant through the perturbing acoustic signal (i.e., Va = 0), then Eq. 7.9 reduces to a standard propagating wave equation with wave speed [{\/M)dP/dp\a\^^^. The earliest theoretical acoustic derivations of wave propagation, developed by Newton, essentially assumed that the temperature is constant as the disturbance travels through the system. Although this may seem like a reasonable approximation, it is incorrect and leads to an incorrect value of the speed of sound. To lowest order, the situation can be further analyzed by examining the energy equation, Eq. 7.4, recalling that we are considering only the ideal fluid with A. = 0. We can then write + V ot
u
= 0
(7.10)
Ly
and use Eq. 7.7 to obtain a r
;^ r
R
'
/?
1
= 0 (7.11) where we have been consistent in keeping terms of lowest order and have assumed that the isochoric heat capacity is constant to this order. The time invariance of the quantity in the brackets means that Tp~^^^^ is constant during the perturbation. Using the ideal gas equation of state (discussed below), this leads to the constancy of Pp-(^v-\-R)/Cv^ which is a common expression for an adiabatic process in an ideal gas. (Recall that the isobaric heat capacity for an ideal gas is given by Cp = Cy -\- R, and the exponent of p is then y, the ratio of heat capacities, which has a value of 5/3 for an ideal monatomic gas [2].) Thus, at least in this limit, the perturbing force leads to an adiabatic process which implies that the entropy is constant during passage of an acoustic disturbance. This turns out generally to be a good approximation, and we choose entropy, S, for the variable a in Eq. 7.9. With V5 = 0, we can write the wave equation VV = 0
(7.12)
s to represent the hydrodynamic effect of a small disturbance on a fluid in equilibrium. In common with the interpretation of other wave equations in
242
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
physical systems, we conclude that the perturbation propagates unchanged with a propagation speed given by
^ = W T 7 ^
(7.13)
Equation 7.13 works remarkably well for a wide variety of fluid acoustic situations despite the approximations made in its derivation. This common expression for speed of sound is often not used directly, because derivatives at constant entropy are not often tabulated. Using a standard definition of the adiabatic compressibility, KS = {l/p)dp/dP\s, we can write w^ =
(7.14) MpKs
while additional standard thermodynamic relations allow us to write w' =
Cp
dP
MCy
dp
(7.15)
The relations derived from the Helmholtz free energy then give us the expression in Eq. 7.1. We conclude that a small-amplitude disturbance in an equilibrium ideal fluid will propagate as an isentropic wave with wave speed given by Eq. 7.1; this is a sound wave. 7.2.2 The Speed of Sound In statistical mechanics, an ideal gas is often considered to be a system of noninteracting point particles; the monatomic noble gases at low densities approximate this ideal condition. In this case, the equation of state can be written as P = p/^T, the molar isochoric heat capacity is generated from the three translational degrees of freedom as 3JR/2, and the isobaric heat capacity is 5RI2. For this system, Eq. 7.15 immediately gives a value of 322.59 m/s for the speed of sound in argon (M = 0.039 948 kg/mol) at 300 K. A currently accepted reference value at atmospheric pressure is 322.67 m/s, from which we can conclude that this simple theory and model works remarkably well. (Note that this degree of agreement between the properties of a real fluid and the ideal gas model should not generally be expected even at 1 atm, especially for heavier molecules.) Returning to our basic Eq. 7.1, note that the Helmholtz free energy for such a monatomic ideal gas can be written as A^>(p, T) = RT[A^-^\np
- 3/2 In J ]
(7.16)
WAVE EQUATION IN A FLUID
243
where A^ is an integration constant witli which we need not be concerned [4]. With this equation for A, the derivatives required for Eq. 7.1 are easily calculable with the result w' =
^-^ (7.17) 3M as previously seen for an ideal monatomic fluid. More generally, an ideal gas can have additional (intramolecular) degrees of freedom and internal structure. This is reflected in different values or temperature dependences in the ideal gas heat capacities of structured molecules, or, equivalently, in additional temperature dependence in the expression for the ideal gas Helmholtz free energy. With estimates for the translational, rotational, vibrational, and electronic energies based on spectroscopic information, the ideal gas heat capacities, using models as the rigid-rotor harmonic oscillator, can be calculated from statistical mechanics. From this information, values for the speed of sound of low-density systems can be calculated. Alternatively, accurate measurements of the speed of sound at low density can be used to provide estimates of the ideal gas properties of a molecular system. (However, if the frequency associated with the sound is not commensurate with the relaxation time of the modes contributing to the ideal gas calculations, corrections must be considered when connecting measured sound speeds to ideal gas behavior [5].) Interactions between the particles in a fluid (intermolecular effects) are extremely important when considering the speed of sound, or any other thermodynamic properties, outside the limited region in which the ideal gas approximation is vaUd. In this case, Eq. 7.16 is not an appropriate representation of the fluid's Helmholtz free energy, and Eq. 7.17 cannot be used to determine the speed of sound. Equations 7.1 and 7.13 through 7.15 are appropriate for compressed gases at elevated pressures, for saturated or compressed liquids, and for supercritical fluids. Explicit examples for the use of these equations in determining the speed of sound in a variety of fluids and over a range of state variables are given Section 7.4. Two general classes of approximations have been made in our derivation of the wave equation and the equivalent speed of sound expressions of Eqs. 7.1 and 7.13 through 7.15. These require that (1) the ampUtude of the wave be small so that the equations can be linearized, and (2) reversibility must be maintained so that isentropic behavior can be assumed. We shall briefly explore these requirements in this Section. The small-amplitude condition is easily fulfilled in most common examples of sound propagation. The restriction can be made that the maximum relative deviation of the density, the acoustic "condensation," is very small, \P - Po\/Po < < 1; alternatively, one can require that \P — Po\ << MpoW^. This condition can be satisfied if the pressure perturbations are much smaller than the equilibrium value of the pressure, which is typically the case.
244
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
To further quantify the small-ampHtude restriction, we note that the soundintensity level can be defined by a logarithmic ratio of an average energy flux in a propagating wave relative to a reference value for such a flux; SIL = 10 log(///ref), where SIL is the sound-intensity level expressed in decibels referenced to /ref. (Typically the reference value is 10~^^ W/m^ for sound in air, although common practices for this arbitrary reference depend on the application; e.g., for certain underwater acoustics applications, 6.76 x 10~^^ W/m^, equivalent to an acoustic pressure of about 1 jiPa, may be used.) The energy flux for a plane wave can be approximated as W^M(PA — Po)'^/(2po), where PA is the maximum value (or amplitude) of the density in the wave. In air at ambient conditions, the small-amplitude approximation breaks down (i.e., \P — Po\/Po = 1) at a sound-intensity level (re 10~^^ W/m^) of about 194 decibels (dB); for comparison, the range of hearing at 1000 Hz is about 0 to 140 dB — so the small-amplitude restriction holds for sounds with an energy of some 250,000 times that which may damage the human ear. For liquid water at ambient conditions, this condition would hold up to about 242 dB (re 10"^^ W/m^); with the reference based on 1 |iPa, this would be about 304 dB. We can conclude that, except in the cases of extraordinarily large "shock" pressures, the linear approximations hold for sound propagation. The more significant restriction on our derivation of the sound equation is associated with the irreversibility caused by the viscous and heat flux effects that were ignored despite their importance in Eqs. 7.3 and 7.4. Additionally, the inability of a fluid to instantaneously return to an equilibrium state when perturbed can be related to the relaxation times that govern the fluid's behavior. The energy in a molecular system is distributed among internal (such as vibrational and rotational modes) and translational degrees of freedom; when the time scales for populating these energy levels are not compatible with the frequency of the propagating sound, the derivations given previously will not be accurate. We will not go into details here, but these effects lead to both attenuation of sound and dispersion — the dependence of the speed of sound on the frequency. One can make various approximations to the full hydrodynamic equations to obtain corrections to the speed of sound equations given previously. The equations can be solved in the complex plane, and to lowest order, the viscous and thermal effects are found to leave the speed of sound unchanged, and contribute only to a frequency-dependent absorption [6]. More generally, one can obtain an expansion of the form w^(f) = w^{0)[l + aX + 0(X^)]
(7.18)
for the real part of the speed of sound as a function of the frequency, / ; the expansion parameter, X, is related to the relaxation times and the frequency.
EQUATIONS OF STATE FOR FLUIDS
245
and the coefficient a can be evaluated for certain models or treated as an empirical parameter. The solution given by Eq. 7.1 represents the zero frequency limit, and, depending on the particular relaxation mechanisms present, some dispersion effects may be seen at frequencies above 1 GHz. The frequency at which dispersion is important for a given system depends on the rotational and vibrational relaxation times in the molecule; these are generally related to the molecular symmetry. Note that the high-frequency signals correspond to small wavelengths, and it is at small wavelengths that the continuum approximation, essential in the development of hydrodynamic approaches to the fluid state, is inappropriate. Finally, we note that the fluctuation-driven phenomena and the density correlations present in near-critical fluids will have an effect on the isentropic assumption and may contribute to dispersion at even lower frequencies. Details concerning these aspects of the acoustic approximation can be found in the texts listed in "Additional Reading."
7.3 Equations of State for Fluids From the previous discussion, we conclude that the speed of sound in a fluid can be determined from its Helmholtz free-energy surface; see Eq. 7.1. In general, an expression for A is formed from the sum of ideal gas and residual contributions: A{pT) = A'^(p, T) + A'*(p, T). The ideal-gas term represents energy associated with noninteracting particles and their internal degrees of freedom—i.e., from the statistical partition function involved with translational, electronic, vibrational, and rotational motions. This quantity can be computed from models of the molecular architecture (i.e., rigid-rotator, harmonic oscillator model for nonlinear polyatomic molecules) and spectroscopic data. Following the format of Eq. 7.16, the ideal gas portion of the Helmholtz energy can be written in general as A^HP, T) = RT[A^ + Inp + f(T)]
(7.19)
where / gives all of the temperature dependence. It is often sufficient to consider only the A^^ contribution for gas phase systems at very low density; however, the residual contribution becomes important as the pressure is increased. The residual contribution to A describes the effects of interparticle interactions. An equation of state, typically an expression for the pressure as a function of temperature and density (or volume), can also be decomposed into the ideal gas and residual contributions. The ideal-gas term in the equation of state is known theoretically as P^^ = pRT, and it is the ideal gas heat capacity (or related thermodynamic quantity) that is used to obtain A'^. Accurate expressions for the residual contributions to A or P are generally based on empirical
246
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
thermodynamic data. The same decomposition of the Helmholtz free energy into ideal and residual parts holds also for mixtures; in this case, the ideal gas contribution is simply related to a mole fraction average of the pure components' ideal gas values, but cross-interaction effects make the residual values more difficult to determine. In Section 7.4, we see how these two parts of A contribute to the speed of sound in fluids. In Figures 7.1 and 7.2 we sketch the basic phase topology of simple fluid systems to understand the scope of the Helmholtz energy formulation for fluid systems. All pure fluids have phase diagrams similar to Figure 7.1; however, quantum fluids may look somewhat different at low temperatures, and some fluids may decompose at temperatures within the area depicted. The solid-fluid phase boundary intersects the vapor-liquid equilibrium (VLB) phase boundary at the triple point, a single value of temperature and pressure at which the solid, liquid, and vapor phases coexist. The melting line extends, in principle, to infinite pressures (although there are, of course, other phenomena at extreme conditions), but the VLB phase boundary terminates at a critical point. For temperatures above the critical-point temperature, the fluid is said to be in the supercritical phase; for lower temperatures, the fluid is a vapor at pressures below the VLB phase boundary and a liquid at higher pressures (to the solidfluid boundary). Along the VLB phase boundary, the fluid is said to be in the saturated liquid or saturated vapor state; inside the two-phase region of
Critical point
Supercritical fluid
Temperature
FIG. 7.1. Pressure vs temperature phase diagram of a simple pure fluid. SLE is solidliquid equilibrium and VLB is vapor-liquid equilibrium.
24'
EQUATIONS OF STATE FOR FLUIDS
Density FIG. 7.2. Pressure vs density phase diagram for a simple pure fluid. Figure 7.2, the equilibrium state consists of two phases, and the speed of sound is not uniquely defined. The phase topology for mixtures can become significantly more complex. In general, for a system at fixed composition, the VLE phase boundary is no longer a simple line. The liquid will vaporize at a bubble point, and the bubbles may have a composition different from that of the coexisting liquid. The vapor condenses at a dew point, and the coexisting liquid may also have a different composition. The locus of dew points and bubble points for a fixed composition may be joined by a critical point. In addition, there may be liquid-liquid immiscibility in portions of the phase diagram. The speed of sound is well defined for each one-phase point, and the same equations given previously can be used for mixtures. 7.3.1 Simple Equations of State There are several ways to represent the residual portion of the thermodynamic surfaces of fluids. The virial series is a density expansion for low densities based on statistical-mechanics cluster approaches [4]. The pressure can be written as P = pRT[\ + B{T)p + C{T)p^ + . . . = pRT[\ + ^
Bi{T)p']
(7.20)
In this notation, B is the second virial coefficient, but Bi is the (/ + 1) virial coefficient. Although the second and, perhaps, third virial coefficients can
248
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
be determined from experimental data, virial coefficients of higher order are difficult to measure, and the series itself does not converge as the critical density is approached. Equation 7.20 is rigorous in the low density limit, and for gases at low and moderate density, up to about one third of the critical density, this type of representation can be quite accurate. Equation 7.20 yields (7.21) and, using Eq. 7.1, the speed of sound for the virial equation of state is given by
w^ =
M
RT + RTY^{i + \)Bip' (=1
R+2RTY.^,(^
+ B}jp^ + RT^Y.i,i=x{^ + B}j{^ + B^p'+i
(7.22) where primes and double primes indicate the first and second temperature derivatives, respectively, of the virial coefficients, Bi and the temperature term in A'^, / from Eq. 7.19. It is straightforward but tedious to expand the denominator in Eq. 7.22 to write a density series of the form 2 ^ry" w = M
l + ^/^Kr)p'
(7.23)
1=1
where y" is the ratio of the ideal gas heat capacities, 1 — ( 2 7 / + T^/") in the current notation. The first two terms are given by [7] '
KxiJ) = 2Bi + liy" - l)TBi +
(1 - y " ) ^
T "
^T^B^
(7.24)
and y" - 1 K2iT) = ^—^[Bi yo + ^ yo
, T „ -, 2y'' + 1 + ily" - \)TB, - (1 - Y'')T^B,f + - ^yO— B 2 TB. + ^
-T^B. 2y°
(7.25)
\
EQUATIONS OF STATE FOR FLUIDS
249
Thus, if the virial coefficients are known, they can be used to calculate the speed of sound in a gas. In these expressions, we see the intimate connection between the ideal gas value of the speed of sound and the full expression for this property. In Eq. 7.23, the ideal gas contribution is identified with the zero-density value, w^^ = (RTy^/My^^, but note that a contribution from internal degrees of freedom, y^, also affects the higher order terms in the expression for the speed of sound as seen in Eqs. 7.24 and 7.25. It is generally true that the ideal gas speed of sound is consistent with the low-density limit of a thermodynamic expression, and further that the internal modes in a molecular system also affect the sound speed at high densities. The family of cubic equations of state represents another simple form to describe the thermodynamic surface. In general these equations can be written as [8] pRT ap^ (7.26) P = \ —bp \-\- ubp -\- vb^p^ where u and w serve to distinguish between the type of cubic equation (e.g., van der Waals, Peng-Robinson, etc.), b is typically a fluid dependent constant, and a is also fluid dependent, but may be constant or a function of temperature, depending on the model being considered. The residual Helmholtz free energy can then be written as A'(p,T)
=
-RTln(l-bp)
2-j-bp(u-\-^/u^ -4v) In p\lu^ — Av 2-\-bp{u — yfu^ Av) (7.27)
and the speed of sound becomes w
ap(2-\- bup) [vb^p^ + bup + \f
RT M L (l-bp)^
\_R L1 - ^P 2Rf
-\- RTf"
a'p 1' vb'^p'^ + bup-{-\\ 2-hbp(u-^y/u^ -4v) In b^u^ - 4u ' " 2 -h bp{u H- y/u^ - Av)
(7.28)
The primes again indicate temperature derivatives. The expression simplifies considerably for the original van der Waals equation (where u = v = 0 and a is constant), but the general expression of Eq. 7.28 can be used for any of the common cubic equations of state. (Note that in the case of w = w; = 0, the logarithmic expression can be expanded, and the full equation is still well defined.)
250
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Cubic equations are often used to describe fluids in engineering applications. These equations can provide reasonably good descriptions of a thermodynamic surface, (except near the critical point), and liquid-vapor phase boundary information can be calculated directly from this approach. The sound speeds calculated from expressions such as that given previously may be adequate approximations. For more accurate descriptions of the speed of sound (and other properties), the reference quality equations, as discussed in Section 7.3.2, are required. In addition to these simple equations that can be used to approximate the properties of real fluid systems, we mention that there are also model systems for which properties can be calculated. For instance, the hard-sphere fluid, consisting of (structureless) spherical particles of diameter a and relative mass M, interacts with a potential function which is infinite when the particles collide and zero otherwise. This model forms the basis for many computer simulations and theoretical results and is often applied to real systems. The equation of state for this fluid has been shown empirically to be well described by the Camahan-Starling expression [9] P = pRT
^ ; / \3 (1 - r]y
'
(7.29)
where rj = 7ta^pNA/6 and NA is the Avogadro number. This leads to a Helmholtz energy expression of the form A^/^rJA^+lnp-^lnr + l ^ ^ ^ j
(7.30)
and Eq. 7.1 immediately gives 2
RT \5 + lOr]- 3rf- - 24rj^ + 31rr' - 22r]^ + 5??^ 1
" = 3M 1
(T^^
)
^'-''^
Unfortunately, we are not aware of any computer simulations which have generated the speed of sound of the hard-sphere system, but the model can be used for real fluids with an effective hard-sphere diameter regressed from available data. Of course, alternative ideal gas behavior can also be incorporated into such a hard-sphere model for real fluids. 7.3.2 Other Equation of State Descriptions The most accurate equations of state are those that seek to represent all available thermodynamic data to within their experimental uncertainty. These are generally expressed by the pressure as a function of temperature and density together with a correlation for the ideal gas properties, or directly
EQUATIONS OF STATE FOR FLUIDS
251
as the Helmholtz free energy as a function of temperature and density. Such equations are available for industrially important fluids and are suitable for purposes such as instrument calibration and custody transfer applications. These equations are developed by regression of large numbers of data over a broad range of state variables and may have 30 to 40 adjustable coefficients to describe the vapor, liquid, and supercritical phases, including the phase boundary information. Of course there are also thermodynamic surfaces of intermediate complexity compared with the simple cubic or virial coefficient representations and the reference quality equations being considered here. Examples of the accurate reference quality equations of state include the Benedict-Webb-Rubin family of equations, which have the form of P = pRT + J^ Gip'^^T'^^ + Yl GiP'^'T'^' exp(}/p2) i=\
(7.32)
i=Ni
Typical equations of this form [10] have 32 coefficients G/, with the exponents of density ranging from 2 to 13 and the exponents of temperature ranging from —4 to ^; several important fluids have been described by this functional form with fixed exponents, nt and m,, by regressing only the coefficients Gt based on the available experimental data on density, heat capacity, phase boundary, speed of sound, etc. This type of equation can also be converted into an expression for the Helmholtz free energy, and the speed of sound can be calculated from Eq. 7.1 as before. Recently, several equations of the form A(p, T) = A'^{p, r ) + ^
G.-^'r'^' + 2
i=\
G,(5"'T'"'- exp((5^0
i=Ni
N
+ X]G,^"'^'"'exp[g(5,r)]
(7.33)
i=N2
have been developed for important fluids, where 8 = pjPc, t = r ^ / r , the subscript "c" indicates a value at the critical point, and ^ is a function that improves the description of the thermodynamic surface in the critical region. These equations are also determined empirically from the best available experimental data; in this case, the numbers of terms in each sum, the exponents «/, m/, //, and the function g may be optimized from the data. The resultant "structurally optimized" thermodynamic surface generally provides the best general representation over the full range of fluid states, and the speed of sound determined using Eqs. 7.1 and 7.33 provide an accurate estimate of the property. The sounds speed tabulated in Section 7.6 were generated from models such as those of Eqs. 7.32 and 7.33.
252
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
When reference-quality thermodynamic surfaces are not available for a fluid of interest, predictive models for the surface can be used to determine the speed of sound. The residual Helmholtz energy surfaces of similar fluids generally obey a corresponding states principle [4, 8]; thus, with two-scale factors, representing changes in energy and distance scales, one can generate an approximate surface for a fluid based on the thermodynamic surface of a more well-studied fluid. This approach allows reasonable estimates of the speed of sound with a minimal amount of information. The cubic equations of state of Eq. 7.26 can be implemented when values for the critical temperature and pressure are known; the more complex cubic equations require additional information, and other extensions of the corresponding states principle can be implemented that make use of available experimental information. The molecular structure must be considered explicitly to approximate the ideal-gas effects, which are quite significant; group additivity and related methods can be useful for this problem when spectroscopic data are not available. Direct predictions of the heat capacities and compressibilities can also be made; these enter into the speed of sound calculation using Eq. 7.15. Alternatively, corresponding states or related models can be developed for the speed of sound itself, and these can be considered independently of the full Helmholtz energy surface. Fluid mixtures generally obey the same thermodynamic and hydrodynamic principles as pure fluids, and thus equations of state can be constructed that look similar to those discussed previously, and the speed of sound can be computed using the same equations, with derivatives being taken at fixed composition. Among the special considerations, we note that the continuity equation, Eq. 7.2, must be revised to reflect conservation of each species present, and the irreversibilities associated with demixing upon passage of a sound perturbation must be considered at high frequencies. In addition, as mentioned previously, the phase topology of mixtures generally differs from that of pure fluids, and care must be taken with any calculation of the speed of sound that the appropriate phase has been identified. Thermodynamic surfaces for mixtures can be calculated from a virial equation of state if the pure fluid coefficients are known and the relevant "cross" virial coefficients, representing interactions between the distinct species in the fluid mixture, can be estimated. Cubic equations of state can also be extended to mixtures using well-defined mixing rules and, if available, a set of interaction parameters representing the cross-species effects. Reference quality equations can also be extended to mixtures with appropriate mixing rules; available experimental data over a range of compositions can be used to regress additional correction terms and hence improve the mixture calculations. As a simple, but explicit, example of a calculation of the speed of sound in a mixture, we consider a system at very low densities, so that the ideal
EQUATIONS OF STATE FOR FLUIDS
253
gas equations are appropriate. The equation of state, P = pRT, remains valid for ideal gas mixtures, and we denote the isobaric and isochoric ideal gas heat capacities of a mixture as C^p^^^ and Cy j^^^. Equation 7.15 can be expressed as
^y^'^f^^la^IL V
mix/
ig
yi^ .
(7.34) V
/
^V,mix ^^"^"^^^
Note again that this is the same as for a pure fluid, e.g. Eq. 7.17, although in that case, we considered a monatomic gas with heat capacities 3R/2 and 5R/2. The pure components of the mixture (denoted by a subscript /) more generally have ideal gas heat capacities C^- and C^ •, and they are present with a mole fraction of JC/. It is rigorously correct that the molar mass and ideal gas heat capacities of a mixture can be written as mole-fraction-weighted averages of these quantities for the pure components: Mmix = S-^/M; C^p,mix — ^^i^^p,i''> C(f jj^.^ = Ejc/Cif.. Then Eq. 7.34 can be written as
From Eq. 7.35 we see that the speed of sound of a mixture, even in an ideal gas state, cannot generally be calculated as a mole fraction average of the speed of sound of the components. Near the critical point of a pure fluid or a mixture, none of the equations of state discussed earlier is adequate. At this singularity in the thermodynamic surface, the behavior is governed by fluctuations that have very long length scales rather than by the comparatively short-ranged interactions observed in other regions of the phase diagram [11]. The phenomenon of "critical slowing" affects the hydrodynamics of a near-critical system, so the derivation of the speed of sound as a thermodynamic property must be reexamined [7]. The thermodynamic speed of sound is strictly zero at the critical point, which can be seen by Eq. 7.15 with the additional observation that the strong critical divergence of the isobaric heat capacity is compensated by a similar decrease in the isothermal derivative; thus, the weak divergence of the isochoric heat capacity in the denominator of the expression ensures that the speed of sound approaches zero at the critical point. This value of the speed of sound can be obtained from a classical Helmholtz free energy surface, although a correct description of the approach at the critical point requires a nonanalytical, scaling theory expression for the free energy that is outside the scope of this chapter. The dispersion of sound (i.e., the dependence of the speed of sound on frequency) is very pronounced in the critical region, and it is the low-frequency limit that is governed by these thermodynamic considerations.
254
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
In summary, the thermodynamic speed of sound of a fluid system at any temperature, density (or pressure), and composition can be obtained by using Eq. 7.1 if the Helmholtz free energy surface for the fluid is known. The quality of the speed of sound calculated in this manner of course depends on the quality of the thermodynamic surface that is used. The reference quality equations can reliably be used, as they should reflect the uncertainty of the available speed of sound data.
7.4 Speed Of Sound in Fluids Argon is one of the simplest fluid systems because of its location on the periodic table, and it is also an important commodity chemical. The system has been well studied, and the reference quality Helmholtz energy equation of Tegeler et al. [12] provides an excellent basis for examining the behavior of the speed of sound in argon. The equation of [12] is of the form described by Eq. 7.33; in addition, a modified Benedict-Webb-Rubin equation as in Eq. 7.32 is available in [10]. In Figures 7.3 to 7.5, we show the behavior of the speed of sound in various projections over a large range of the state variables. These and other figures are based on the sound speed calculations available in the NIST pure fluids database [3]. In Figure 7.3, the saturation boundary is shown in bold; at a Argon
80
380
680
Temperature, K
FIG. 7.3. The speed of sound in argon vs temperature along isobars. The heavy lines give values on the saturated liquid and saturated vapor curves.
255
SPEED OF SOUND IN FLUIDS
Argon 1000
30 40 Pressure, MPa FIG. 7.4. The speed of sound in argon vs pressure along isotherms. The heavy Hnes give values on the saturated liquid and saturated vapor curves.
Argon
185
235 285 Temperature, K
335
385
FIG. 7.5. The speed of sound in argon vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves.
256
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Argon
0.03 0.02 0.01
I
0.00
CO
!'#lM'iS
-0.01 H
'
I
I
250
300
350
tr'
-0.02 -I -0.03
100
150
200
400
450
Temperature, K
FIG. 7.6. Sample deviations between experimental and calculated values for the speed of sound in argon. Two points (with deviations of —0.26% and 0.54%) are not shown. fixed temperature, the two lines represent the sound speed in the coexisting vapor and liquid. Note that the sound speeds approach zero at the critical point. Note from Figures 7.3 and 7.4 that the speed of sound isotherms cross in these pressure projections. The isochores (lines of constant density) are also seen to cross in Figure 7.5; for supercritical fluids, the speed of sound tends to increase with density and with temperature along the isochores. In Figure 7.6, we compare the calculated sound speeds with the data of Estrada-Alexanders and Trusler [13]. The 173 data were obtain between 110 K and 450 K for pressures ranging from 6.83 kPa to 19.26 MPa using a spherical acoustic resonator with experimental uncertainties from 0.001 to 0.007%. These data were used in the regression of the thermodynamic surface, and the deviations between the data and this reference quality surface are consistent with the experimental uncertainty assessment. In general, as discussed previously the reference quality equations of state provide calculations of the speed of sound that are consistent with the experimental data. In Figure 7.7, we compare the 300 K sound speed isotherm for argon as calculated from the equation of state of [12] with the results of various models. A density of 30 mol dm~^ for this temperature corresponds to a pressure of about 190 MPa. The dashed line gives the ideal gas value for argon at 300 K, as calculated from Eq. 7.17. This quantity is independent of density, and agrees quite well with the reference equation of state at zero density. In fact, all of the models agree with Eq. 7.17 in this limit. We have included
257
SPEED OF SOUND IN FLUIDS
Argon: 300 K
Virial: 3PE feng Robinson
Virial: 2PE
0
5
10
15
20
25
30
35
Density, mol dm~^
FIG. 7.7. Speed of sound of argon vs density at 300 K. The bold line labeled EOS represents values calculated from the standard reference equation of state. The other lines are values calculated from various models discussed in the text. three lines to represent the virial expressions of Eqs. 7.20 to 7.25. The line labeled "Virial: 2E0S" used the first two acoustic virial coefficients [Eqs. 7.24 and 7.25], where the pressure virial coefficients and their derivatives were calculated numerically from a computer implementation of the reference thermodynamic surface of Tegeler et al. [12]. The line "Virial: 2PE" used the first two acoustic virial coefficients tabulated by Estrada-Alexanders and Trusler [13] that had been determined from an interatomic potential energy function. The differences in these two lines come about because of the difficulty in numerically determining the second temperature derivatives of the pressure virial coefficients and the sensitivity of the acoustic virial coefficients to these quantities. The model ("Virial: 3PE"), using three virial coefficients as tabulated by Estrada-Alexanders and Trusler [13], is seen to provide the best representation of the reference line even to the high pressures which are shown. The van der Waals and Peng-Robinson equations of state are two forms in the family of cubic equations of state. The parameters of both of these equations are determined from the critical parameters of argon and, in the case of the Peng-Robinson equation, the acentric factor for argon. For the van der Waals equations, both u and v of Eq. 7.28 are zero; for the Peng-Robinson equation of state w = 2 and u = — 1. These equations may have problems reproducing experimental densities at high values of the pressure, and the
258
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
speeds of sound are seen to deviate significantly from the reference line. Finally, Figure 7.7 shows the results for the speed of sound of the hard sphere fluid as given in Eq. 7.31. Only a value of the effective hard sphere diameter, a is needed in order to evaluated the parameter rj. We have determined a = 2.88 X 10~^^ m by requiring that the hard sphere result be consistent with the sound speed of argon at 20 mol dm~^. The hard sphere expression seems to give somewhat less curvature than the reference line for this isotherm. Nitrogen provides our second example, and in Figures 7.8 to 7.10 we indicate the behavior of the speed of sound based on the Helmholtz energy formulation of Span et al. [14]. The general behaviors of the curves shown are seen to be quit similar to those of argon; for similar state points, nitrogen generally has a larger value of the speed of sound. Figure 7.11 compares the calculated speed of sound with high-quality experimental measurements on argon of Costa Gomes and Trusler [5]. The data were obtained in a spherical resonator operating at frequencies between 5 kHz and 26 kHz, and the experimental uncertainties were estimated [5] as less than 0.01%. A difference between the measured results at these frequencies and the thermodynamic speed of sound, corresponding to zero frequency, was noted by Costa Gomes and Trusler. Figure 7.11 also shows the data as adjusted to zero frequency in [5]. These zero-frequency data were considered in the regression of Span
Nitrogen 2000-1
1800 J
500 MPa
1600-^ CO
E
1400 J
c 1200 H o w 1000H 'o "D 800 H 0
600-^ 400 200 0
T
1
1
1
1
1
T
110 160 210 260 310 360 410 460 510 560 Temperature, K
FIG. 7.8. The speed of sound in nitrogen vs temperature along isobars. The heavy lines give values on the saturated liquid and saturated vapor curves.
259
SPEED OF SOUND IN FLUIDS
Nitrogen 100 K
900 n
0 0
5" 300 H
I
10
20
T 1 r 30 40 50 Pressure, MPa
r 60
T 70
"I 80
FIG. 7.9. The speed of sound in nitrogen vs pressure along isotherms. The heavy Unes give values on the saturated liquid and saturated vapor curves.
Nitrogen lOOOn
225 275 Temperature, K
FIG. 7.10. The speed of sound in nitrogen vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves.
260
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Nitrogen
0.01 0.00 -0.01
m
* la i
+AA SAA
I I
^
i
-0.02 -0.03 -0.04
'•••-.
-0.05 10
15
20
25
30
Pressure, MPa
FIG. 7.11. Sample deviations between experimental and calculated values for the speed of sound in nitrogen. The closed symbols represent data obtained at frequencies between 5 kHz and 26 kHz; the open symbols are data corrected to zero frequency. Circles: 250 K; squares: 275 K; triangles: 300 K; diamonds: 350 K. et al. [14] and, as can be seen, the reference equation of state agrees with the data to within their uncertainty. The properties of water are well described by the standard formulation adopted by the International Association for the Properties of Water and Steam (lAPWS) [15]. Figures 7.12 to 7.14 show the behavior of the speed of sound of fluid water. Because of the relatively high critical temperature of water (647.096 K) relative to that of argon and nitrogen, the region of phase equilibria is expanded in these figures. The vertical line segments in Figures 7.12 and 7.13 connected the isobars and isotherms, respectively, across the two-phase region. This is the region of liquid-vapor phase separation, and the thermodynamic speed of sound is not defined for the two-phase systems. Figure 7.12 includes a line representing the critical isobar (Pc = 22.064 MPa); this isobar intersects the phase boundary at the critical temperature, whereas all subcritical isobars meet the saturation boundary at liquid and vapor points, and supercritical isobars do not intersect the phase boundary. As mentioned earlier the thermodynamic speed of sound should vanish at the critical point, however, the classical equation for water gives a small but non-zero value. For some equations of state, this value may depend on the computer implementation and the particular calculation performed: a value of about 35 m/s can be obtained on the phase boundary at the critical temperature for the lAPWS equation.
261
SPEED OF SOUND IN FLUIDS
Water
500 MPa
Temperature, K
FIG. 7.12. The speed of sound in water vs temperature along isobars. The heavy lines give values on the saturated liquid and saturated vapor curves. Water
1600-1
400 K
1400 HV
275 K
300 K 500 K
w 1200-^
1 1000 H o CO
° 800-^ ^
600 H
700 K _^
Nw
400-]
200-1
1
1
1
1
0
2
4
6
8 10 12 14 16 18 20 22 24
1
1
1
1
1
1
1—r~
Pressure, MPa
FIG. 7.13. The speed of sound in water vs pressure along isotherms. The heavy lines give values on the saturated liquid and saturated vapor curves.
262
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Several features of the behavior of the speed of sound in water can be observed in Figure 7.13. Note that there is a small region where the speed of sound in the saturated vapor (the lower branch of the phase boundary in most of the figures) is greater than that in the coexisting liquid. This occurs for temperatures between about 643 K and the critical temperature. Near the pure-fluid critical point, the vapor and liquid states become quite similar; although the vapor has a smaller density, the speed of sound in the vapor is larger than that of the liquid in this region. The subcritical isotherms in Figure 7.13 illustrate another feature: note that for fixed pressure (at about 18 MPa, to be specific), the speed of sound at 300 K and 400 K is significantly higher than that at 275 K, and the 500 K isotherm is inverted compared to the trend of the others. This behavior is related to the maxima seen in the isobars in Figure 7.12. Figure 7.14 shows the clear maximum of the speed of sound in the saturated liquid and the broader maximum in that saturated vapor phase. The behavior of the speed of sound in air is shown in Figures 7.15 to 7.17. These values were calculated from the thermodynamic surface of Lemmon et al. [16] which provides a reference quality equation of state for dry air considered at the fixed composition of 0.7812 nitrogen, 0.2096 oxygen, and 0.0092 argon by mole fraction. We see that the general behavior of the mixture isobars (Fig. 7.15), isotherms (Fig. 7.16), and isochores (Fig. 7.17) is quite similar to the trends seen for the pure fluids. Note that the subcritical isobar at 0.1 MPa also shows a segment in the two-phase region. This line is Water 1600
275
375
475
575
675
775
T 875
1175
Temperature, K
FIG. 7.14. The speed of sound in water vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves.
263
SPEED OF SOUND IN FLUIDS
Air 1150
950
•D C 13 O (0
750
550 •D
0 0 Q.
350 H
150 60
160
260
360 Temperature, K
460
560
FIG. 7.15. The speed of sound in air vs temperature along isobars. The heavy Unes give values on the saturated liquid and saturated vapor curves (bubble-point curve and dew-point curve).
Air
30 Pressure, MPa
FIG. 7.16. The speed of sound in air vs pressure along isotherms. The heavy hues give values on the saturated liquid and saturated vapor curves (bubble-point curve and dew-point curve).
264
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Air »uu20 mol dm-3
700^
|600-
10 mol dm~3 5 mol dm'
§500o CO
O400•D 0) CD
(g-300-
0.1 mol
^ ~ \ ^ ^
200100-
60
1
1
11
,
,
,
,
110
160
210
260
310
360
410
T
460
1
1
510
560
Temperature
FIG. 7.17. The speed of sound in air vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves (bubble-point curve and dew-point curve). not vertical, however, and does not connect states in equilibrium. The lower saturation boundary is the line of dew points. At 0.1 MPa, the dew-point temperature for the composition specified is at about 81.6 K. The bubble-point temperature is at about 78.8 K. The bubbles in equilibrium with the bulk liquid and the condensate in equilibrium with the bulk vapor have different compositions than the bulk fluid. Again, the thermodynamic speed of sound is not defined in the two-phase region. Finally, in Figure 7.18 we show some deviations between experimental sound speeds for a natural gas fluid and calculations from a model for the mixture Helmholtz free energy. The data were obtained with a cylindrical cavity for a methane-rich ten-component mixture representing Gulf Coast natural gas [17]. The experimental uncertainty is about 0.05%. The generalized mixture model representing the base line [18] uses the reference quality equation of state for each pure component and a few mixture parameters based primarily on density and phase equilibrium data. The model is seen to agree with the data to nearly within the experimental uncertainty.
7.5 Conclusions The thermodynamic properties of a fluid system are completely determined by the Helmholtz free energy surface of the system. We have seen that the
265
CONCLUSIONS
Natural gas 0.14 0.12 -I
0.1 •] ^
0.08 \
c •B 0.06 CO
Q
•
0.04 A 0.02
• • • " •
•
— •
0 H -0.02 4
6 Pressure, MPa
8
10
12
FIG. 7.18. Sample deviations between experimental and calculated values for the speed of sound in a natural gas. speed of sound in such a system can be considered a thermodynamic property, and that it can be calculated from Eq. 7.1 The available sources for the speed of sound in fluid systems can be considered to be of three types: (1) experimental measurements and compilations of experimental measurements can provide the most accurate values for the speed of sound in vv^ell-defined systems; (2) reference quality Helmholtz free energy surfaces have been developed for important fluids and can describe the speed of sound essentially to within the uncertainty of available data; (3) models or predictive schemes can be used for systems for which measurements are not available, including more complex molecular systems and mixtures. Among the sources for experimental data, the tables [19] of the NIST Thermodynamics Research Center provide an extensive compilation for both hydrocarbon and non-hydrocarbon systems. These tables are available in both printed and electronic form. Researchers at the National Institute of Standards and Technology have developed several reference thermodynamic surfaces based on experimental measurements at NIST and elsewhere, and have compiled reference quality surfaces from several other sources. Such information is available in the form of PC-based computer programs such as the NIST Thermophysical Properties of Pure Fluids Database [3]. NIST also provides predictive computer packages that can be used for pure fluids with limited measurements and for mixtures. The catalog of computer packages from the Standard Reference Data
266
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
office of NIST is available on-line at http://www.mst.gov/srd. Some information on the thermophysical properties of fluids is available in the NIST chemistry webbook at http://webbook.nist.gov. The material in Tables 7.1 through 7.4 was calculated from these PC computer programs.
7.6 Tables of the Speed of Sound in Important Fluids TABLE 7.1. Argon — Saturation Boundary Vapor
Liquid
r(K) 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
P (MPa)
p (mol/dm-^)
CO (m/s)
p (mol/dm-^)
0) (m/s)
0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.15 0.16 0.18 0.19 0.21 0.23 0.25 0.28 0.30 0.32 0.35 0.38 0.41 0.44 0.47 0.51 0.54 0.58 0.62 0.67 0.71 0.76 0.81 0.86 0.91 0.97 1.02 1.08 1.15 1.21 1.28 1.35 1.43 1.50 1.58 1.66 1.75 1.84
35.44 35.28 35.13 34.98 34.82 34.67 34.51 34.35 34.19 34.04 33.87 33.71 33.55 33.39 33.22 33.05 32.89 32.72 32.54 32.37 32.20 32.02 31.84 31.66 31.48 31.30 31.11 30.92 30.73 30.54 30.34 30.14 29.94 29.74 29.53 29.32 29.11 28.89 28.67 28.45 28.22 27.98 27.75 27.50 27.25
861.10 854.24 847.35 840.43 833.47 826.48 819.45 812.39 805.28 798.14 790.95 783.72 776.45 769.14 761.78 754.37 746.91 739.40 731.83 724.21 716.54 708.81 701.02 693.16 685.24 677.26 669.20 661.08 652.87 644.60 636.24 627.79 619.26 610.64 601.93 593.11 584.19 575.16 566.02 556.75 547.36 537.83 528.16 518.34 508.36
0.104 0.115 0.127 0.140 0.155 0.170 0.186 0.204 0.222 0.242 0.264 0.286 0.310 0.336 0.363 0.392 0.422 0.454 0.488 0.524 0.562 0.601 0.643 0.687 0.733 0.782 0.833 0.887 0.943 1.002 1.065 1.130 1.198 1.269 1.344 1.423 1.506 1.592 1.683 1.778 1.878 1.982 2.092 2.208 2.330
168.28 169.08 169.87 170.64 171.39 172.12 172.83 173.52 174.20 174.85 175.49 176.11 176.71 177.29 177.85 178.39 178.91 179.41 179.89 180.36 180.80 181.22 181.62 182.01 182.37 182.71 183.03 183.33 183.61 183.87 184.11 184.33 184.52 184.70 184.85 184.98 185.09 185.17 185.23 185.27 185.29 185.28 185.24 185.18 185.10
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.1.
(continued) Vapor
Liquid
T(K)
P (MPa)
p (mol/dm^)
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
1.93 2.03 2.12 2.23 2.33 2.44 2.55 2.67 2.79 2.91 3.04 3.17 3.30 3.44 3.59 3.74 3.89 4.05 4.21 4.38 4.55 4.73
27.00 26.74 26.47 26.20 25.91 25.62 25.32 25.01 24.68 24.35 23.99 23.62 23.23 22.82 22.38 21.90 21.38 20.81 20.16 19.40 18.45 17.03
TABLE 7.2. T (K) 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340
267
CO (m/s)
498.21 487.88 477.36 466.63 455.69 444.51 433.10 421.42 409.47 397.21 384.62 371.63 358.17 344.14 329.41 313.80 297.06 278.88 258.79 236.08 209.33 174.74
p (mol/dm^)
CO (m/s)
2.458 2.592 2.735 2.885 3.043 3.211 3.389 3.579 3.781 3.996 4.228 4.477 4.747 5.041 5.363 5.719 6.119 6.574 7.106 7.750 8.587 9.875
184.98 184.85 184.68 184.49 184.27 184.02 183.74 183.41 183.03 182.60 182.09 181.50 180.81 180.00 179.05 177.93 176.57 174.89 172.74 169.81 165.39 157.01
Argon —Isobar at 0.101 325 MPa p (mol/dm^)
Q) (m/s)
0.140 0.125 0.113 0.103 0.095 0.088 0.082 0.077 0.072 0.068 0.064 0.061 0.058 0.056 0.053 0.051 0.049 0.047 0.045 0.044 0.042 0.041 0.039 0.038 0.037 0.036
173.81 184.14 193.74 202.78 211.38 219.59 227.49 235.09 242.44 249.56 256.48 263.21 269.76 276.15 282.39 288.50 294.48 300.33 306.07 311.70 317.24 322.67 328.02 333.27 338.45 343.55 (continues)
268
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.2. T (K) 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
(continued)
p (moydm^) 0.035 0.034 0.033 0.032 0.031 0.030 0.030 0.029 0.028 0.028 0.027 0.026 0.026 0.025 0.025 0.024 0.024 0.023 0.023 0.023 0.022 0.022 0.021 0.021 0.021 0.020 0.020 0.020 0.019 0.019 0.019 0.018 0.018 0.018 0.018 0.017
TABLE 7.3. T (K) 90 100 110 116.5981 116.5981
120 130 140 150 160
co (m/s) 348.57 353.52 358.40 363.21 367.96 372.65 377.28 381.86 386.38 390.85 395.27 399.63 403.95 408.23 412.46 416.65 420.79 424.90 428.96 432.99 436.98 440.93 444.85 448.73 452.58 456.40 460.19 463.94 467.67 471.36 475.03 478.67 482.28 485.87 489.42 492.96
Argon — Isobar at 1 MPa p (mol/dm^)
(o (m/s)
34.579 32.955 31.157 29.821 1.240 1.182 1.046 0.945 0.865 0.799
824.17 751.59 672.31 614.12 184.63 189.35 201.51 212.09 221.67 230.54
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.3. (continued) T (K)
170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
p (mol/dm^)
(o (m/s)
0.744 0.697 0.656 0.619 0.587 0.558 0.532 0.509 0.488 0.468 0.450 0.433 0.418 0.403 0.390 0.377 0.366 0.355 0.344 0.335 0.325 0.317 0.309 0.301 0.293 0.286 0.280 0.273 0.267 0.261 0.256 0.250 0.245 0.240 0.235 0.231 0.226 0.222 0.218 0.214 0.211 0.207 0.203 0.200 0.197 0.194 0.190 0.187 0.185 0.182 0.179 0.176 0.174 0.171
238.88 246.79 254.34 261.59 268.58 275.34 281.89 288.26 294.47 300.52 306.44 312.22 317.88 323.44 328.88 334.23 339.49 344.66 349.74 354.75 359.68 364.54 369.33 374.06 378.72 383.33 387.87 392.36 396.80 401.19 405.52 409.81 414.06 418.25 422.41 426.52 430.59 434.63 438.62 442.58 446.50 450.39 454.24 458.06 461.85 465.60 469.33 473.02 476.69 480.33 483.94 487.52 491.08 494.61
269
270
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.4. T (K)
90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620
Argon — Isobar at 5 MPa p (mol/dm-^)
CO (m/s)
34.879 33.343 31.682 29.828 27.655 24.831 19.159 6.090 4.947 4.307 3.867 3.534 3.269 3.051 2.865 2.706 2.566 2.443 2.332 2.232 2.142 2.060 1.984 1.914 1.849 1.789 1.734 1.681 1.632 1.586 1.543 1.502 1.464 1.427 1.392 1.359 1.328 1.298 1.270 1.242 1.216 1.191 1.167 1.145 1.123 1.101 1.081 1.061 1.043 1.024 1.007 0.990 0.973 0.958
844.78 777.34 705.93 628.46 541.00 433.54 248.19 206.39 223.58 236.49 247.45 257.20 266.11 274.40 282.20 289.60 296.66 303.44 309.98 316.29 322.42 328.37 334.17 339.82 345.34 350.74 356.03 361.22 366.31 371.31 376.23 381.06 385.82 390.51 395.13 399.69 404.18 408.62 412.99 417.32 421.59 425.81 429.99 434.12 438.20 442.25 446.25 450.21 454.13 458.02 461.87 465.68 469.46 473.21
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.4. (continued) T (K)
630 640 650 660 670 680 690 700
p (mol/dm^) 0.942 0.928 0.913 0.899 0.886 0.873 0.860 0.848
CO (m/s) 476.92 480.60 484.26 487.88 491 Al 495.04 498.58 502.09
TABLE 7.5. Argon —Isobar at 10 MPa T (K)
90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460
p (mol/dm^)
0) (m/s)
35.23 33.78 32.25 30.59 28.76 26.67 24.15 20.87 16.44 12.31 9.89 8.45 7.49 6.78 6.23 5.79 5.42 5.11 4.83 4.60 4.38 4.20 4.02 3.87 3.73 3.60 3.48 3.37 3.26 3.17 3.07 2.99 2.91 2.83 2.76 2.70 2.63 2.57
868.40 805.91 741.46 674.25 603.33 527.43 445.10 357.76 284.47 259.27 260.68 267.70 275.72 283.74 291.51 298.98 306.16 313.06 319.72 326.15 332.37 338.42 344.29 350.02 355.60 361.05 366.38 371.60 376.72 381.74 386.67 391.51 396.27 400.96 405.58 410.12 414.60 419.02 (continues)
271
272
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.5. T (K) 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
TABLE 7.6. T (K)
100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310
(continued)
p (mol/dm^) 2.51 2.46 2.41 2.36 2.31 2.26 2.22 2.18 2.13 2.10 2.06 2.02 1.99 1.95 1.92 1.89 1.86 1.83 1.80 1.77 1.75 1.72 1.70 1.67
o) (m/s) 423.38 427.69 431.94 436.14 440.29 444.39 448.45 452.46 456.43 460.36 464.25 468.10 471.91 475.69 479.44 483.15 486.83 490.47 494.09 497.67 501.23 504.76 508.26 511.73
Argon — Isobar at 50 MPa p (mol/dm^)
(0 (m/s)
36.26 35.19 34.12 33.05 31.97 30.90 29.83 28.76 27.70 26.65 25.63 24.62 23.65 22.71 21.81 20.96 20.14 19.38 18.66 17.98 17.34 16.75
969.74 927.77 887.55 849.02 812.28 777.44 744.66 714.08 685.82 659.98 636.63 615.80 597.47 581.56 567.93 556.40 546.77 538.85 532.42 527.30 523.31 520.31
TABLES O F T H E SPEED O F SOUND IN IMPORTANT FLUIDS
TABLE 7.6. T (K)
320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
TABLE 7.7. T (K) 110 120 130 140 150 160 170
(continued)
p (mol/dm^)
(0 (m/s)
16.19 15.67 15.18 14.72 14.28 13.88 13.50 13.14 12.80 12.48 12.17 11.88 11.61 11.35 11.10 10.87 10.64 10.43 10.22 10.02 9.83 9.65 9.48 9.31 9.15 9.00 8.85 8.71 8.57 8.43 8.30 8.18 8.06 7.94 7.83 7.72 7.61 7.51 7.41
518.16 516.73 515.91 515.62 515.78 516.32 517.19 518.35 519.74 521.33 523.11 525.03 527.09 529.26 531.52 533.87 536.29 538.77 541.31 543.89 546.50 549.15 551.83 554.54 557.26 560.00 562.75 565.51 568.28 571.05 573.83 576.62 579.40 582.18 584.97 587.75 590.52 593.30 596.07
. •
Argon —Isobar at 100 MPa p (mol/dm-^)
0) (m/s)
37.41 36.57 35.74 34.93 34.13 33.35 32.58
1074.29 1043.43 1014.38 987.00 961.21 936.96 914.20 (continues)
273
274
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.7. T (K) 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
(continued)
p (mol/dm^)
co (m/s)
31.83 31.09 30.37 29.66 28.98 28.31 27.66 27.03 26.42 25.82 25.25 24.69 24.15 23.63 23.13 22.65 22.18 21.73 21.30 20.88 20.47 20.09 19.71 19.35 19.00 18.66 18.34 18.02 17.72 17.43 17.15 16.87 16.61 16.35 16.10 15.86 15.63 15.40 15.19 14.97 14.77 14.57 14.37 14.18 14.00 13.82 13.65 13.48 13.31 13.15 13.00 12.84 12.70
892.89 873.00 854.48 837.28 821.35 806.65 793.12 780.71 769.35 759.00 749.60 741.08 733.39 726.46 720.25 714.70 709.76 705.38 701.52 698.13 695.17 692.61 690.42 688.56 687.02 685.75 684.74 683.98 683.43 683.08 682.91 682.92 683.08 683.38 683.81 684.37 685.03 685.80 686.67 687.62 688.65 689.76 690.94 692.17 693.47 694.83 696.23 697.68 699.17 700.71 702.28 703.88 705.52
275
TABLES O F T H E SPEED O F SOUND IN IMPORTANT FLUIDS
TABLE 7.8.
Nitrogen — Saturation Boundary Vapor
Liquid T (K)
64 65 66 67 68 69 70 71 72 73 74 75 76 11 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115
P (MPa)
p (mol/dm^)
(i) (m/s)
p (mol/dm-^)
(D (m/s)
0.015 0.017 0.021 0.024 0.028 0.033 0.039 0.045 0.051 0.059 0.067 0.076 0.086 0.097 0.109 0.122 0.137 0.153 0.169 0.188 0.208 0.229 0.252 0.276 0.303 0.331 0.360 0.392 0.426 0.462 0.500 0.541 0.583 0.628 0.676 0.726 0.778 0.834 0.892 0.953 1.016 1.083 1.153 1.226 1.303 1.383 1.466 1.553 1.643 1.737 1.835 1.937
30.83 30.69 30.54 30.39 30.24 30.09 29.93 29.78 29.62 29.47 29.31 29.15 28.99 28.83 28.67 28.51 28.34 28.17 28.01 27.84 27.66 27.49 27.32 27.14 26.96 26.78 26.60 26.41 26.22 26.03 25.84 25.64 25.44 25.24 25.03 24.82 24.61 24.39 24.17 23.94 23.71 23.47 23.23 22.98 22.72 22.46 22.18 21.90 21.61 21.31 20.99 20.66
986.57 976.36 966.18 956.04 945.93 935.83 925.74 915.66 905.58 895.49 885.39 875.28 865.15 855.00 844.82 834.61 824.36 814.07 803.74 793.36 782.93 772.44 761.89 751.28 740.60 729.84 719.01 708.09 697.09 685.99 674.80 663.50 652.09 640.57 628.92 617.14 605.23 593.17 580.96 568.58 556.03 543.30 530.37 517.24 503.88 490.29 476.44 462.32 447.92 433.19 418.12 402.67
0.028 0.033 0.038 0.044 0.051 0.059 0.068 0.077 0.088 0.100 0.112 0.126 0.142 0.158 0.176 0.196 0.217 0.240 0.265 0.292 0.320 0.351 0.383 0.418 0.456 0.496 0.538 0.584 0.632 0.683 0.737 0.795 0.856 0.921 0.990 1.063 1.141 1.223 1.310 1.403 1.501 1.605 1.716 1.833 1.958 2.092 2.234 2.386 2.549 2.724 2.912 3.116
162.08 163.20 164.30 165.37 166.41 167.43 168.42 169.39 170.32 171.23 172.10 172.95 173.77 174.55 175.31 176.03 176.72 177.38 178.00 178.60 179.16 179.68 180.17 180.63 181.05 181.43 181.78 182.10 182.38 182.62 182.82 182.99 183.12 183.21 183.26 183.28 183.25 183.18 183.08 182.93 182.74 182.51 182.24 181.93 181.58 181.19 180.76 180.28 179.75 179.15 178.49 177.75 (continues)
276
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.8.
{continued)
Liquid T (K)
116 117 118 119 120 121 122 123 124 125 126
P (MPa)
p (mol/dm-^)
2.043 2.153 2.268 2.387 2.511 2.639 2.773 2.912 3.056 3.207 3.365
20.31 19.94 19.55 19.13 18.68 18.19 17.63 17.00 16.23 15.21 13.28
TABLE 7.9.
Vapor CO (m/s) 386.80 370.43 353.49 335.85 317.33 297.68 276.54 253.32 227.00 195.48 150.97
p (mol/dm-^)
0) (m/s)
3.337 3.579 3.844 4.137 4.465 4.838 5.270 5.785 6.430 7.324 9.111
176.93 176.01 175.00 173.87
Nitrogen —Isobar at 0.101 325 MPa
T (K)
p (mol/dm^)
(D (m/s)
70
29.94 28.77 0.1646 0.1392 0.1125 0.0946 0.0817 0.0720 0.0643 0.0581 0.0530 0.0488 0.0452 0.0420 0.0393 0.0369 0.0348 0.0329 0.0312 0.0297 0.0283 0.0271 0.0259 0.0249 0.0239 0.0230 0.0221 0.0214 0.0206 0.0200 0.0193 0.0187 0.0182
926.19 851.39 174.82 190.42 212.07 231.36 248.99 265.37 280.74 295.28 309.11 322.33 335.02 347.22 359.00 370.38 381.40 392.08 402.44 412.51 422.31 431.84 441.12 450.16 458.99 467.60 476.02 484.25 492.31 500.20 507.93 515.51 522.95
77.355 77.355
90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670
\11.6\ 171.17 169.49 167.43 164.67 160.26 148.38
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.9. T (K) 690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990 1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730
(continued)
p (moydm^) 0.0177 0.0172 0.0167 0.0162 0.0158 0.0154 0.0150 0.0147 0.0143 0.0140 0.0137 0.0134 0.0131 0.0128 0.0126 0.0123 0.0121 0.0118 0.0116 0.0114 0.0112 0.0110 0.0108 0.0106 0.0104 0.0102 0.0101 0.0099 0.0097 0.0096 0.0094 0.0093 0.0092 0.0090 0.0089 0.0088 0.0086 0.0085 0.0084 0.0083 0.0082 0.0081 0.0080 0.0079 0.0078 0.0077 0.0076 0.0075 0.0074 0.0073 0.0072 0.0071 0.0070
co (m/s) 530.27 537.45 544.53 551.49 558.34 565.10 571.76 578.33 584.82 591.22 597.55 603.80 609.99 616.10 622.15 628.14 634.07 639.94 645.75 651.51 657.22 662.87 668.48 674.04 679.56 685.03 690.45 695.84 701.18 706.48 711.75 716.97 722.16 727.31 732.43 737.51 742.56 747.57 752.56 757.51 762.43 767.31 772.17 777.00 781.80 786.57 791.31 796.03 800.72 805.38 810.02 814.63 819.21 (continues)
277
278
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
TABLE 7.9. (continued) T (K)
p (mol/dm^) 0.0070 0.0069 0.0068 0.0067 0.0067 0.0066 0.0065 0.0064 0.0064 0.0063 0.0062 0.0062 0.0061
1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
CO (m/s) 823.78 828.31 832.82 837.31 841.78 846.22 850.64 855.04 859.42 863.77 868.10 872.42 876.71
TABLE 7.10. Nitrogen — Isobar at 1 MPa T (K)
p (mol/dm^)
0) (m/s)
70 90
30.00 26.69 23.77 1.48 1.32 1.02 0.8523 0.7359 0.6498 0.5829 0.5291 0.4848 0.4475 0.4158 0.3883 0.3643 0.3432 0.3244 0.3076 0.2925 0.2788 0.2663 0.2549 0.2445 0.2349 0.2260 0.2178 0.2101 0.2030 0.1963 0.1901 0.1843
932.51 727.06 559.22 182.79 194.09 221.53 243.20 261.98 278.91 294.53 309.13 322.92 336.03 348.56 360.58 372.16 383.33 394.13 404.58 414.73 424.57 434.15 443.46 452.53 461.37 469.99 478.42 486.65 494.70 502.58 510.30 517.88
103.747 103.747
110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.10.
(continued)
T (K)
p (mol/dm^)
(0 (m/s)
670 690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990
0.1788 0.1736 0.1687 0.1641 0.1597 0.1556 0.1516 0.1479 0.1443 0.1410 0.1377 0.1346 0.1317 0.1289 0.1262 0.1236 0.1211 0.1187 0.1164 0.1142 0.1120 0.1100 0.1080 0.1061 0.1043 0.1025 0.1008 0.0991 0.0975 0.0959 0.0944 0.0930 0.0916 0.0902 0.0888 0.0876 0.0863 0.0851 0.0839 0.0827 0.0816 0.0805 0.0795 0.0784 0.0774 0.0764 0.0755 0.0745 0.0736 0.0727 0.0719 0.0710 0.0702
525.31 532.61 539.79 546.84 553.79 560.63 567.37 574.01 580.57 587.04 593.43 599.74 605.98 612.14 618.24 624.27 630.25 636.16 642.01 647.81 653.55 659.24 664.89 670.48 676.02 681.52 686.98 692.39 697.76 703.09 708.38 713.63 718.84 724.02 729.16 734.26 739.33 744.37 749.37 754.34 759.28 764.18 769.06 773.91 778.72 783.51 788.27 793.00 797.71 802.39 807.04 811.67 816.27
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710
(continues)
279
280
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.10. T (K) 1730 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990 2010
TABLE 7.11.
(continued)
p (mol/dnr) 0.0694 0.0686 0.0678 0.0670 0.0663 0.0656 0.0649 0.0642 0.0635 0.0628 0.0622 0.0616 0.0609 0.0603 0.0597
CO (m/s)
820.84 825.39 829.92 834.42 838.90 843.36 847.79 852.20 856.59 860.96 865.31 869.63 873.94 878.22 882.48
Nitrogen — Isobar at 5 MPa
T (K)
p (mol/dm^)
CO (m/s)
70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650
30.283 27.202 23.385 16.433 6.029 4.387 3.605 3.108 2.753 2.483 2.266 2.089 1.940 1.812 1.702 1.605 1.519 1.442 1.373 1.311 1.254 1.202 1.154 1.111 1.070 1.032 0.997 0.965 0.934 0.906
959.16 772.11 565.77 288.81 226.09 254.67 277.24 296.46 313.52 329.07 343.47 356.96 369.71 381.84 393.42 404.54 415.24 425.55 435.52 445.18 454.54 463.63 472.48 481.09 489.48 497.67 505.67 513.49 521.15 528.65
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.11.
(continued)
T (K)
p (mol/dwr)
0) (m/s)
670 690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990
0.879 0.853 0.830 0.807 0.786 0.765 0.746 0.728 0.711 0.694 0.678 0.663 0.649 0.635 0.622 0.609 0.597 0.585 0.574 0.563 0.553 0.543 0.533 0.524 0.515 0.506 0.498 0.490 0.482 0.474 0.467 0.460 0.453 0.446 0.439 0.433 0.427 0.421 0.415 0.409 0.404 0.399 0.393 0.388 0.383 0.378 0.374 0.369 0.365 0.360 0.356 0.352 0.348
536.01 543.24 550.33 557.31 564.17 570.93 577.59 584.15 590.62 597.01 603.32 609.55 615.70 621.79 627.81 633.77 639.66 645.49 651.27 657.00 662.67 668.29 673.86 679.38 684.86 690.29 695.68 701.02 706.33 711.60 716.82 722.01 727.16 732.28 737.36 742.40 747.42 752.40 757.34 762.26 767.14 772.00 776.82 781.62 786.38 791.12 795.83 800.52 805.17 809.81 814.41 818.99 823.55
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710
(continues)
281
282
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.11.
(continued)
T (K)
p (mol/dm^)
1730 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
0.344 0.340 0.336 0.332 0.329 0.325 0.322 0.318 0.315 0.312 0.309 0.305 0.302 0.299
TABLE 7.12.
CO (m/s) 828.08 832.59 837.07 841.53 845.97 850.38 854.78 859.15 863.50 867.82 872.13 876.42 880.68 884.93
Nitrogen —Isobar at 10 MPa
T (K)
p (mol/dm^)
0) (m/s)
70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670
30.60 27.75 24.49 20.41 14.96 10.28 7.87 6.53 5.65 5.02 4.54 4.15 3.84 3.57 3.35 3.15 2.98 2.82 2.69 2.56 2.45 2.35 2.26 2.17 2.09 2.02 1.95 1.89 1.83 1.77 1.72
989.62 819.44 647.61 472.34 331.25 292.70 300.47 315.42 330.91 345.77 359.83 373.13 385.75 397.78 409.27 420.29 430.88 441.09 450.94 460.47 469.71 478.67 487.38 495.85 504.10 512.15 520.01 527.69 535.21 542.57 549.80
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.12. T (K)
690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990 1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730
(continued)
p (mol/dm-^) 1.67 1.62 1.58 1.54 1.50 1.46 1.43 1.39 1.36 1.33 1.30 1.27 1.25 1.22 1.20 1.17 1.15 1.13 1.11 1.09 1.07 1.05 1.03 1.01 1.00 0.981 0.965 0.950 0.935 0.920 0.906 0.893 0.880 0.867 0.855 0.843 0.831 0.820 0.809 0.798 0.787 0.777 0.767 0.758 0.748 0.739 0.730 0.721 0.713 0.704 0.696 0.688 0.680
(D (m/s) 556.88 563.84 570.69 577.42 584.05 590.58 597.01 603.36 609.63 615.82 621.93 627.97 633.95 639.86 645.71 651.49 657.23 662.90 668.53 674.10 679.62 685.10 690.53 695.92 701.26 706.56 711.82 717.05 722.23 727.37 732.48 737.56 742.59 747.60 752.57 757.51 762.42 767.29 772.14 776.96 781.74 786.50 791.23 795.93 800.61 805.26 809.88 814.48 819.05 823.60 828.12 832.62 837.10 {continues)
283
284
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.12. T (K) 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
TABLE 7.13.
(continued)
p (mol/dm^) 0.673 0.665 0.658 0.651 0.644 0.637 0.630 0.624 0.617 0.611 0.605 0.599 0.593
CO (m/s) 841.55 845.98 850.38 854.77 859.13 863.47 867.79 872.09 876.37 880.63 884.87 889.09 893.29
Nitrogen —Isobar at 50 MPa
T (K)
p (mol/dm^)
(0 (m/s)
90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670 690 710
30.52 28.54 26.61 24.76 22.99 21.35 19.83 18.45 17.22 16.12 15.14 14.28 13.50 12.81 12.19 11.64 11.13 10.67 10.25 9.87 9.51 9.18 8.88 8.60 8.33 8.08 7.85 7.64 7.43 7.24 7.05 6.88
1061.02 963.85 878.83 806.60 747.47 701.00 665.92 640.44 622.64 610.73 603.25 599.08 597.38 597.50 598.98 601.45 604.67 608.44 612.61 617.08 621.78 626.63 631.59 636.63 641.73 646.85 651.99 657.14 662.28 667.41 672.53 677.63
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.13. (continued) T (K)
p (mol/dm^)
(0 (m/s)
730 750 770 790 810 830 850 870 890 910 930 950 970 990
6.72 6.56 6.41 6.27 6.14 6.01 5.88 5.77 5.65 5.54 5.44 5.34 5.24 5.15 5.06 4.97 4.89 4.81 4.73 4.66 4.59 4.51 4.45 4.38 4.32 4.25 4.19 4.13 4.08 4.02 3.97 3.92 3.87 3.82 3.77 3.72 3.67 3.63 3.59 3.54 3.50 3.46 3.42 3.38 3.34 3.31 3.27 3.24 3.20 3.17 3.13 3.10 3.07
682.70 687.76 692.79 697.80 702.78 707.74 712.67 717.58 722.46 727.31 732.14 736.95 741.73 746.49 751.22 755.93 760.62 765.28 769.92 774.54 779.14 783.71 788.26 792.79 797.30 801.79 806.26 810.70 815.13 819.54 823.92 828.29 832.64 836.96 841.27 845.56 849.83 854.09 858.32 862.54 866.73 870.91 875.08 879.22 883.35 887.46 891.56 895.63 899.70 903.74 907.77 911.78 915.78
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730 1750 1770
(continues)
285
286
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.13.
(continued)
T (K)
p (mol/dm-^)
CO (m/s)
1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
3.04 3.01 2.98 2.95 2.92 2.89 2.86 2.84 2.81 2.78 2.76
919.76 923.73 927.68 931.61 935.53 939.44 943.33 947.20 951.06 954.91 958.74
TABLE 7.14. T (K)
90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670 690 710 730 750
Nitrogen —at 100 MPa p (mol/dm^)
CO (m/s)
32.57 31.03 29.57 28.19 26.90 25.69 24.57 23.52 22.54 21.63 20.78 19.99 19.26 18.58 17.94 17.35 16.79 16.27 15.79 15.33 14.90 14.49 14.11 13.75 13.41 13.09 12.78 12.49 12.21 11.95 11.69 11.45 11.22 11.00
1244.55 1173.11 1110.14 1055.27 1008.09 967.97 934.20 906.03 882.74 863.64 848.09 835.56 825.55 817.66 811.52 806.84 803.39 800.95 799.36 798.49 798.21 798.44 799.10 800.12 801.46 803.07 804.91 806.95 809.16 811.53 814.04 816.67 819.40 822.23
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.14. (continued) T (K)
p (mol/dm^)
CO (m/s)
770 790 810 830 850 870 890 910 930 950 970 990
10.79 10.58 10.39 10.20 10.02 9.85 9.68 9.52 9.36 9.21 9.07 8.93 8.79 8.66 8.53 8.41 8.29 8.17 8.06 7.95 7.84 7.74 7.64 7.54 7.44 7.35 7.26 7.17 7.08 7.00 6.92 6.84 6.76 6.68 6.61 6.54 6.46 6.39 6.33 6.26 6.19 6.13 6.07 6.01 5.95 5.89 5.83 5.77 5.72 5.66 5.61 5.56 5.51
825.14 828.12 831.18 834.29 837.46 840.68 843.94 847.25 850.58 853.95 857.35 860.78 864.23 867.69 871.18 874.68 878.20 881.73 885.27 888.82 892.38 895.95 899.52 903.10 906.68 910.26 913.84 917.43 921.01 924.60 928.18 931.77 935.35 938.93 942.50 946.07 949.64 953.21 956.76 960.32 963.87 967.41 970.95 974.48 978.01 981.53 985.04 988.55 992.05 995.55 999.03 1002.51 1005.98
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730 1750 1770 1790 1810
(continues)
287
288
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.14.
(continued)
T (K)
p (mol/dm^)
CO (m/s] 1
1830 1850 1870 1890 1910 1930 1950 1970 1990
5.46 5.41 5.36 5.31 5.27 5.22 5.18 5.13 5.09
1009.45 1012.91 1016.36 1019.8C1 1023.23 1026.66 1030.08 1033.491 1036.9C»
T A B L E 7.15.
Water—Saturation Boundary Vapor
Liquid T (K)
275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343
P (MPa)
p (mol/dm^)
0) (m/s)
p (mol/dm^)
0.0007 0.0008 0.0009 0.0011 0.0012 0.0014 0.0016 0.0018 0.0020 0.0023 0.0026 0.0030 0.0033 0.0038 0.0042 0.0047 0.0053 0.0059 0.0066 0.0073 0.0081 0.0090 0.0100 0.0111 0.0123 0.0135 0.0149 0.0164 0.0180 0.0198 0.0217 0.0238 0.0260 0.0284 0.031
55.50 55.50 55.50 55.50 55.49 55.48 55.47 55.45 55.43 55.41 55.38 55.36 55.33 55.30 55.27 55.23 55.20 55.16 55.12 55.08 55.03 54.99 54.94 54.90 54.85 54.80 54.74 54.69 54.63 54.58 54.52 54.46 54.40 54.34 54.28
1411.35 1420.78 1429.78 1438.35 1446.51 1454.29 1461.68 1468.71 1475.39 1481.72 1487.72 1493.41 1498.78 1503.86 1508.64 1513.14 1517.37 1521.33 1525.04 1528.49 1531.70 1534.68 1537.42 1539.95 1542.25 1544.35 1546.24 1547.93 1549.43 1550.73 1551.85 1552.80 1553.56 1554.16 1554.59
0.0003 0.0003 0.0004 0.0005 0.0005 0.0006 0.0007 0.0008 0.0008 0.0010 0.0011 0.0012 0.0013 0.0015 0.0017 0.0019 0.0021 0.0023 0.0026 0.0028 0.0031 0.0034 0.0038 0.0042 0.0046 0.0050 0.0055 0.0060 0.0066 0.0072 0.0078 0.0085 0.0093 0.010 0.011
CO (m/s)
410.33 411.77 413.21 414.64 416.06 417.48 418.89 420.29 421.69 423.08 424.46 425.84 427.21 428.57 429.93 431.28 432.62 433.96 435.29 436.61 437.93 439.24 440.54 441.83 443.11 444.39 445.66 446.92 448.17 449.41 450.64 451.87 453.08 454.28 455.48
289
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.15. (continued) Vapor
Liquid T (K)
345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447
P (MPa) 0.034 0.037 0.040 0.043 0.047 0.051 0.055 0.060 0.065 0.070 0.075 0.081 0.087 0.094 0.101 0.108 0.116 0.125 0.133 0.143 0.153 0.163 0.174 0.186 0.198 0.211 0.224 0.238 0.253 0.269 0.286 0.303 0.321 0.340 0.360 0.381 0.403 0.426 0.449 0.474 0.500 0.527 0.556 0.585 0.616 0.648 0.681 0.716 0.752 0.789 0.828 0.869
p
{mol/dw?) 54.21 54.15 54.08 54.01 53.95 53.88 53.81 53.73 53.66 53.59 53.51 53.44 53.36 53.28 53.20 53.12 53.04 52.96 52.88 52.79 52.71 52.62 52.53 52.45 52.36 52.27 52.18 52.08 51.99 51.90 51.80 51.71 51.61 51.51 51.42 51.32 51.22 51.11 51.01 50.91 50.81 50.70 50.59 50.49 50.38 50.27 50.16 50.05 49.94 49.82 49.71 49.60
0) (m/s)
1554.85 1554.96 1554.91 1554.71 1554.36 1553.86 1553.22 1552.44 1551.52 1550.47 1549.29 1547.98 1546.54 1544.97 1543.29 1541.48 1539.56 1537.52 1535.36 1533.09 1530.71 1528.22 1525.62 1522.92 1520.11 1517.20 1514.18 1511.06 1507.85 1504.53 1501.12 1497.61 1494.00 1490.30 1486.51 1482.62 1478.64 1474.57 1470.41 1466.17 1461.83 1457.40 1452.89 1448.29 1443.60 1438.83 1433.97 1429.03 1424.01 1418.90 1413.71 1408.44
p
{moVdw?) 0.012 0.013 0.014 0.015 0.016 0.017 0.019 0.020 0.022 0.023 0.025 0.027 0.029 0.031 0.033 0.035 0.038 0.040 0.043 0.046 0.049 0.052 0.055 0.058 0.062 0.066 0.070 0.074 0.078 0.083 0.087 0.092 0.098 0.103 0.109 0.115 0.121 0.127 0.134 0.141 0.148 0.156 0.164 0.172 0.180 0.189 0.198 0.208 0.218 0.228 0.239 0.250
0) (m/s)
456.67 457.84 459.01 460.16 461.30 462.44 463.56 464.67 465.77 466.86 467.93 469.00 470.05 471.09 472.12 473.13 474.14 475.12 476.10 477.06 478.01 478.95 479.87 480.77 481.67 482.54 483.41 484.25 485.09 485.91 486.71 487.50 488.27 489.02 489.76 490.49 491.19 491.88 492.56 493.22 493.86 494.48 495.09 495.68 496.25 496.81 497.35 497.87 498.37 498.86 499.32 499.77 (continues)
290
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.15.
(continued)
Liquid T (K)
449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547
P (MPa) 0.911 0.954 0.999 1.05 1.09 1.15 1.20 1.25 1.31 1.36 1.42 1.49 1.55 1.62 1.68 1.75 1.83 1.90 1.98 2.06 2.14 2.23 2.31 2.40 2.50 2.59 2.69 2.79 2.89 3.00 3.11 3.22 3.34 3.46 3.58 3.70 3.83 3.97 4.10 4.24 4.38 4.53 4.68 4.83 4.99 5.15 5.32 5.49 5.66 5.84
Vapor
p (mol/dm-^)
0) (m/s)
p (mol/dm-^)
CO (m/s)
49.48 49.36 49.24 49.13 49.01 48.88 48.76 48.64 48.51 48.39 48.26 48.13 48.01 47.88 47.74 47.61 47.48 47.34 47.21 47.07 46.93 46.79 46.65 46.51 46.36 46.22 46.07 45.92 45.77 45.62 45.47 45.32 45.16 45.00 44.84 44.68 44.52 44.36 44.19 44.02 43.86 43.68 43.51 43.34 43.16 42.98 42.80 42.61 42.43 42.24
1403.08 1397.64 1392.12 1386.52 1380.84 1375.08 1369.24 1363.32 1357.32 1351.24 1345.08 1338.84 1332.53 1326.13 1319.66 1313.10 1306.47 1299.76 1292.98 1286.11 1279.16 1272.14 1265.04 1257.85 1250.59 1243.25 1235.83 1228.33 1220.76 1213.10 1205.36 1197.53 1189.63 1181.65 1173.58 1165.43 1157.19 1148.88 1140.47 1131.98 1123.41 1114.74 1105.99 1097.15 1088.21 1079.19 1070.07 1060.86 1051.55 1042.15
0.261 0.273 0.285 0.298 0.311 0.325 0.339 0.354 0.369 0.385 0.401 0.418 0.435 0.453 0.471 0.491 0.510 0.531 0.552 0.573 0.596 0.619 0.643 0.668 0.693 0.719 0.746 0.774 0.803 0.833 0.863 0.895 0.927 0.961 0.995 1.031 1.068 1.105 1.144 1.185 1.226 1.269 1.313 1.358 1.405 1.453 1.503 1.554 1.607 1.662
500.20 500.62 501.01 501.39 501.74 502.08 502.40 502.70 502.98 503.24 503.49 503.71 503.91 504.09 504.25 504.39 504.51 504.61 504.68 504.74 504.77 504.78 504.77 504.74 504.68 504.60 504.49 504.36 504.21 504.03 503.82 503.59 503.34 503.06 502.75 502.41 502.05 501.65 501.23 500.78 500.30 499.79 499.25 498.68 498.07 497.43 496.76 496.06 495.32 494.54
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.15.
(continued) Vapor
Liquid T (K)
P (MPa)
549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647
6.02 6.21 6.40 6.60 6.80 7.00 7.21 7.43 7.64 7.87 8.10 8.33 8.57 8.81 9.06 9.32 9.58 9.84 10.12 10.39 10.68 10.97 11.26 11.56 11.87 12.19 12.51 12.83 13.17 13.51 13.86 14.21 14.57 14.94 15.32 15.71 16.10 16.50 16.91 17.33 17.75 18.19 18.63 19.09 19.55 20.02 20.51 21.01 21.52 22.04
291
p (mol/dm^)
CO (m/s)
42.05 41.86 41.66 41.46 41.26 41.06 40.85 40.64 40.43 40.21 40.00 39.77 39.55 39.32 39.08 38.85 38.60 38.36 38.11 37.85 37.59 37.32 37.05 36.77 36.49 36.20 35.90 35.59 35.28 34.95 34.62 34.28 33.92 33.56 33.18 32.78 32.37 31.93 31.48 31.00 30.48 29.93 29.33 28.68 27.97 27.17 26.26 25.15 23.59 19.84
1032.65 1023.04 1013.34 1003.53 993.61 983.59 973.46 963.21 952.85 942.38 931.78 921.06 910.22 899.25 888.14 876.90 865.53 854.00 842.34 830.52 818.54 806.40 794.09 781.60 768.93 756.07 743.01 729.72 716.21 702.44 688.40 674.06 659.38 644.33 628.86 612.91 596.41 579.28 561.45 542.82 523.31 502.83 481.38 459.03 435.99 412.53 388.55 362.10 325.47 251.20
p (mol/dm^) 1.718 1.776 1.836 1.898 1.962 2.028 2.096 2.167 2.240 2.315 2.393 2.473 2.556 2.643 2.732 2.824 2.920 3.020 3.123 3.230 3.342 3.458 3.579 3.705 3.836 3.973 4.116 4.265 4.422 4.587 4.760 4.942 5.134 5.337 5.552 5.781 6.025 6.286 6.566 6.869 7.198 7.558 7.956 8.401 8.907 9.496 10.21 11.12 12.46 15.90
CO (m/s)
493.73 492.88 491.99 491.07 490.10 489.10 488.05 486.96 485.83 484.65 483.42 482.15 480.83 479.46 478.04 476.56 475.03 473.44 471.80 470.09 468.32 466.48 464.58 462.60 460.55 458.42 456.21 453.91 451.52 449.04 446.45 443.75 440.93 437.99 434.90 431.67 428.26 424.67 420.87 416.82 412.49 407.83 402.77 397.19 390.93 383.74 375.08 363.84 346.60 285.31
292
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.16. T (K) 275 295 315 335 355 373.12 373.12
375 395 415 435 455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
Water—Isobar at 0.101 325 MPa p (mol/dm-^) 55.51 55.39 55.04 54.52 53.88 53.20 0.0332 0.0330 0.0312 0.0296 0.0282 0.0269 0.0258 0.0247 0.0238 0.0229 0.0220 0.0212 0.0205 0.0199 0.0192 0.0186 0.0181 0.0176 0.0171 0.0166 0.0162 0.0157 0.0153 0.0150 0.0146 0.0143 0.0139 0.0136 0.0133 0.0130 0.0128 0.0125 0.0123 0.0120 0.0118 0.0116 0.0113 0.0111 0.0109 0.0107 0.0106 0.0104 0.0102 0.0100 0.0099 0.0097
co (m/s) 1411.51 1487,89 1531.87 1552.00 1553.96 1543.18 472.18 473.52 487.03 499.70 511.80 523.44 534.70 545.62 556.23 566.58 576.68 586.55 596.21 605.68 614.95 624.05 632.98 641.75 650.37 658.84 667.18 675.39 683.47 691.43 699.27 707.00 714.63 722.16 729.59 736.92 744.17 751.33 758.41 765.41 772.33 779.17 785.95 792.66 799.30 805.88 812.40 818.86 825.26 831.61 837.90 844.14
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.17.
Water—isobar at 1 MPa
T (K)
p (mol/dm^)
275 295 315 335 355 375 395 415 435 453.03 453.03
455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
55.53 55.41 55.06 54.54 53.90 53.15 52.29 51.34 50.28 49.24 0.286 0.284 0.268 0.255 0.243 0.232 0.223 0.214 0.207 0.199 0.193 0.186 0.181 0.175 0.170 0.165 0.161 0.156 0.152 0.148 0.145 0.141 0.138 0.135 0.132 0.129 0.126 0.124 0.121 0.119 0.116 0.114 0.112 0.110 0.108 0.106 0.104 0.102 0.101 0.099 0.097 0.096
0) (m/s)
1412.95 1489.38 1533.44 1553.68 1555.76 1543.41 1519.06 1484.24 1439.85 1392.05 501.02 502.85 518.73 532.48 545.19 557.16 568.56 579.48 590.01 600.2 610.09 619.72 629.11 638.29 647.26 656.05 664.67 673.12 681.42 689.58 697.61 705.51 713.29 720.95 728.50 735.95 743.30 750.56 757.72 764.80 771.80 778.72 785.56 792.33 799.03 805.66 812.22 818.73 825.17 831.56 837.89 844.16
293
294
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.18.
Water—Isobar at 5 MPa
T (K)
p (mol/dm^)
CO (m/s)
55.64 55.51 55.15 54.64 54.00 53.25 52.40 51.46 50.41 49.27 48.02 46.64 45.09 43.35 43.15 1.41 1.30 1.22 1.15 1.09 1.04 1.00 0.956 0.921 0.889 0.859 0.832 0.807 0.784 0.762 0.741 0.722 0.704 0.686 0.670 0.655 0.640 0.626 0.612 0.600 0.587 0.576 0.564 0.554 0.543 0.533 0.524 0.515 0.506 0.497 0.489 0.481
1419.38 1496.03 1540.44 1561.11 1563.72 1552.01 1528.44 1494.56 1451.34 1399.29 1338.56 1268.87 1189.36 1098.15 1087.81 498.04 520.81 540.41 557.18 572.16 585.84 598.56 610.51 621.84 632.65 643.03 653.03 662.71 672.09 681.21 690.09 698.76 707.23 715.52 723.65 731.62 739.45 747.14 754.7 762.15 769.48 776.72 783.85 790.88 797.83 804.69 811.47 818.18 824.81 831.37 837.86 844.29
275 295 315 335 355 375 395 415 435 455 475 495 515 535 537.09 537.09
555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.19.
Water—Isobar at 10 MPa
T (K)
p (mol/dm^)
(D (m/s)
55.78 55.63 55.27 54.76 54.12 53.38 52.54 51.60 50.58 49.45 48.23 46.88 45.38 43.70 41.78 39.47 38.21 3.08 2.85 2.56 2.36 2.21 2.09 1.99 1.90 1.82 1.75 1.69 1.63 1.58 1.53 1.48 1.44 1.40 1.37 1.33 1.30 1.27 1.24 1.21 1.19 1.16 1.14 1.12 1.09 1.07 1.05 1.03 1.02 1.00 0.98 9.95
1427.48 1504.34 1549.15 1570.33 1573.56 1562.62 1539.98 1507.23 1465.38 1415.01 1356.38 1289.39
275 295 315 335 355 375 395 415 435 455 475 495 515 535 555 575 584.15 584.15
595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
nn.Ai 1127.35 1028.37 910.76 847.33 472.51 494.96 524.55 547.42 566.64 583.48 598.63 612.52 625.42 637.54 649.02 659.95 670.42 680.49 690.21 699.62 708.74 717.62 726.27 734.71 742.96 751.03 758.95 766.71 774.33 781.83 789.2 796.46 803.60 810.65 817.60 824.46 831.24 837.93 844.54
295
296
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.20.
Water—Isobar at 50 MPa
T (K)
p (mol/dm^)
0) (m/s)
275 295 315 335 355 375 395 415 435 455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995
56.82 56.59 56.19 55.67 55.05 54.34 53.56 52.70 51.77 50.76 49.69 48.53 47.29 45.96 44.52 42.95 41.23 39.32 37.17 34.70 31.78 28.24 24.09 19.85 16.46 14.09 12.44 11.24 10.34 9.62 9.04 8.55 8.13 7.77 7.45 7.17 6.91 6.68 6.47 6.28 6.10 5.94 5.78 5.64 5.51 5.38 5.26 5.15 5.04 4.94
1494.62 1570.91 1617.39 1641.44 1648.59 1642.61 1626.02 1600.54 1567.36 1527.34 1481.12 1429.22 1372.03 1309.84 1242.87 1171.23 1094.94 1013.97 928.41 838.86 746.93 659.20 590.51 556.44 556.75 571.69 590.63 609.70 627.78 644.65 660.37 675.07 688.87 701.91 714.26 726.03 737.29 748.08 758.46 768.47 778.15 787.53 796.64 805.49 814.11 822.53 830.74 838.77 846.64 854.34
1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.21.
Water—Isobar at 100 MPa
T (K)
p (mol/dm^)
(0 (m/s)
275 295 315 335 355 375 395 415 435 455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995
58.00 57.67 57.24 56.71 56.10 55.43 54.69 53.90 53.05 52.14 51.18 50.17 49.11 47.99 46.81 45.58 44.27 42.90 41.45 39.92 38.31 36.62 34.84 32.99 31.08 29.15 27.24 25.40 23.67 22.08 20.65 19.37 18.24 17.24 16.37 15.59 14.90 14.28 13.72 13.22 12.76 12.35 11.96 11.61 11.28 10.98 10.70 10.44 10.19 9.95
1583.41 1654.36 1699.71 XllASA 1734.86 1732.90 1721.44 1702.11 1676.13 1644.40 1607.67 1566.57 1521.67 1473.50 1422.55 1369.34 1314.37 1258.17 1201.31 1144.39 1088.12 1033.35 981.04 932.30 888.40 850.76 820.40 797.37 781.16 771.04 766.04 765.04 767.02 771.13 776.73 783.33 790.61 798.30 806.26 814.35 822.49 830.64 838.74 846.78 854.73 862.60 870.36 878.01 885.56 893.00
1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
297
298
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.22.
Air—Saturation Boundary
T (K)
P (Bubble) (MPa)
Liquid p (mol/dm^)
CO (m/s)
P (Dew) (MPa)
Vapor p (mol/dm^)
CO (m/s)
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
0.006 0.007 0.008 0.010 0.012 0.014 0.017 0.020 0.024 0.027 0.032 0.037 0.042 0.049 0.056 0.063 0.072 0.081 0.091 0.10 0.11 0.13 0.14 0.16 0.17 0.19 0.21 0.23 0.26 0.28 0.30 0.33 0.36 0.39 0.42 0.46 0.50 0.53 0.57 0.62 0.66 0.71 0.76 0.81 0.87 0.93 0.99 1.05 1.12 1.18 1.26 1.33 1.41 1.49 1.58 1.66 1.75
33.03 32.89 32.74 32.60 32.46 32.31 32.17 32.02 31.87 31.72 31.58 31.43 31.28 31.13 30.97 30.82 30.67 30.51 30.36 30.20 30.04 29.88 29.72 29.56 29.40 29.23 29.07 28.90 28.73 28.56 28.38 28.21 28.03 27.85 27.67 27.49 27.30 27.12 26.92 26.73 26.53 26.33 26.13 25.92 25.71 25.50 25.28 25.06 24.83 24.60 24.36 24.12 23.87 23.61 23.35 23.08 22.80
1028.29 1020.26 1012.16 1004.00 995.77 987.48 979.13 970.72 962.24 953.70 945.10 936.43 927.70 918.90 910.04 901.11 892.11 883.05 873.91 864.71 855.44 846.09 836.67 827.18 817.61 807.96 798.24 788.44 778.56 768.59 758.55 748.42 738.20 727.90 717.51 707.03 696.46 685.80 675.05 664.20 653.26 642.22 631.08 619.84 608.50 597.06 585.51 573.85 562.09 550.21 538.21 526.10 513.86 501.48 488.97 476.31 463.48
0.0026 0.0033 0.0041 0.0051 0.0063 0.0078 0.0094 0.011 0.014 0.016 0.019 0.023 0.027 0.031 0.037 0.042 0.049 0.056 0.064 0.073 0.082 0.093 0.10 0.12 0.13 0.15 0.16 0.18 0.20 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.45 0.49 0.53 0.57 0.61 0.66 0.71 0.76 0.81 0.87 0.93 0.99 1.06 1.13 1.20 1.28 1.35 1.44 1.52 1.61
0.005 0.006 0.008 0.010 0.012 0.014 0.017 0.021 0.024 0.029 0.034 0.039 0.046 0.053 0.061 0.069 0.079 0.090 0.101 0.114 0.128 0.143 0.160 0.178 0.198 0.219 0.241 0.266 0.292 0.320 0.351 0.383 0.417 0.454 0.493 0.535 0.580 0.627 0.677 0.730 0.786 0.846 0.909 0.976 1.047 1.122 1.201 1.285 1.374 1.468 1.568 1.674 1.786 1.905 2.032 2.166 2.310
155.14 156.38 157.61 158.81 159.99 161.16 162.30 163.42 164.53 165.61 166.66 167.70 168.70 169.69 170.65 171.58 172.49 173.37 174.23 175.05 175.85 176.62 177.36 178.07 178.75 179.40 180.02 180.61 181.17 181.69 182.19 182.65 183.08 183.48 183.84 184.17 184.46 184.72 184.95 185.14 185.30 185.42 185.51 185.55 185.57 185.54 185.48 185.38 185.24 185.07 184.85 184.60 184.30 183.97 183.59 183.17 182.71
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.22.
T (K)
117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
P (Bubble) (MPa)
{continued)
Liquid p (mol/dm^)
0) (m/s)
22.51 22.22 21.91 21.59 21.25 20.90 20.53 20.14 19.73 19.28 18.79 18.24 17.62 16.86 15.87 14.20
450.49 437.29 423.89 410.23 396.30 382.04 367.40 352.31 336.67 320.36 303.21 285.00 265.37 243.75 219.07 189.12
1.85 1.95 2.05 2.16 2.27 2.38 2.50 2.62 2.74 2.87 3.01 3.14 3.28 3.43 3.58 3.72
TABLE 7.23.
P (Dew) (MPa)
Vapor p (mol/dm^)
(o (m/s)
2.463 2.626 2.801 2.989 3.191 3.410 3.648 3.908 4.193 4.510 4.865 5.270 5.740 6.307 7.034 8.127
182.21 181.66 181.08 180.45 179.78 179.06 178.31 177.52 176.68 175.81 174.91 173.96 172.98 171.93 170.79 169.40
1.70 1.80 1.90 2.01 2.12 2.23 2.35 2.47 2.59 2.73 2.86 3.01 3.15 3.31 3.47 3.65
Air—Isobar at 0.101 325 MPa
T (K)
p (mol/dm^)
100 200 300 400 500 600 700 800 900
0.124 0.061 0.041 0.030 0.024 0.020 0.017 0.015 0.014 0.012 0.011 0.010 0.0094 0.0087 0.0081 0.0076 0.0072 0.0068 0.0064 0.0061 0.0058 0.0055 0.0053 0.0051 0.0049 0.0047 0.0045 0.0044 0.0042 0.0041 0.0039 0.0038
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200
299
CO (m/s)
198.21 283.45 347.36 400.50 446.40 487.07 523.89 557.85 589.60 619.60 648.15 675.47 701.72 727.03 751.50 775.19 798.19 820.54 842.30 863.51 884.21 904.43 924.20 943.56 962.52 981.10 999.34 1017.25 1034.84 1052.13 1069.14 1085.87 {continues)
300
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.23.
(continued)
T (K)
p (mol/dm-^)
CO (m/s)
3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.0037 0.0036 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0030 0.0029 0.0028 0.0028 0.0027 0.0026 0.0026 0.0025 0.0025 0.0024
1102.35 1118.58 1134.57 1150.34 1165.89 1181.23 1196.37 1211.31 1226.07 1240.65 1255.05 1269.29 1283.36 1297.27 1311.03 1324.65 1338.12 1351.45
TABLE 7.24.
Air—Isobar at 1 MPa
T (K)
p (mol/dm-^)
(o (m/s)
100
26.59 25.23 1.38 0.616 0.402 0.300 0.240 0.200 0.171 0.150 0.133 0.120 0.109 0.100 0.092 0.086 0.080 0.075 0.071 0.067 0.063 0.060 0.057 0.055 0.052 0.050 0.048 0.046 0.044 0.043 0.041 0.040
658.25 582.97 185.23 281.74 348.45 402.33 448.46 489.16 525.96 559.86 591.54 621.47 649.95 677.21 703.40 728.66 753.07 776.72 799.67 821.98 843.70 864.87 885.53 905.72 925.46 944.79 963.72 982.28 1000.49 1018.37 1035.94 1053.21
106.22 108.10
200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.24. (continued) T (K) 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
p (mol/dm^) 0.039 0.038 0.036 0.035 0.034 0.033 0.032 0.032 0.031 0.030 0.029 0.029 0.028 0.027 0.027 0.026 0.026 0.025 0.025 0.024
co (m/s) 1070.20 1086.91 1103.37 1119.59 1135.56 1151.31 1166.84 1182.17 1197.29 1212.22 1226.97 1241.53 1255.92 1270.15 1284.21 1298.11 1311.86 1325.46 1338.92 1352.24
TABLE 7.25. Air—Isobar at 5 MPa T (K)
p (mol/dm^)
0) (m/s)
100 200 300 400 500 600 700 800 900
27.22 3.38 2.02 1.49 1.18 0.983 0.843 0.738 0.657 0.592 0.539 0.494 0.457 0.424 0.396 0.372 0.350 0.331 0.314 0.298 0.284 0.271 0.259 0.249 0.239 0.230 0.221 0.213 0.206
710.56 279.42 355.63 411.69 458.30 498.93 535.45 569.01 600.34 629.93 658.09 685.05 710.97 735.96 760.13 783.56 806.30 828.42 849.96 870.96 891.46 911.50 931.10 950.29 969.10 987.54 1005.64 1023.41 1040.88
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900
(continues)
301
302
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.25.
(continued)
T (K)
p (mol/dm-^)
0) (m/s)
3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.199 0.193 0.187 0.181 0.176 0.171 0.166 0.162 0.158 0.154 0.150 0.146 0.143 0.139 0.136 0.133 0.130 0.127 0.125 0.122 0.120
1058.05 1074.95 1091.57 1107.95 1124.08 1139.98 1155.65 1171.12 1186.37 1201.43 1216.30 1230.98 1245.49 1259.82 1273.99 1288.00 1301.85 1315.55 1329.10 1342.52 1355.79
TABLE 7.26.
Air—Isobar at 10 MPa
T (K)
p (mol/dm^)
0) (m/s)
100 200 300 400 500 600 700 800 900
27.86 7.39 4.04 2.92 2.32 1.93 1.65 1.45 1.29 1.17 1.06 0.975 0.902 0.839 0.784 0.736 0.693 0.655 0.621 0.591 0.563 0.538 0.515 0.494 0.474 0.456 0.440 0.424 0.410
763.47 296.30 369.50 425.59 471.81 511.90 547.83 580.83 611.64 640.74 668.46 695.01 720.56 745.21 769.07 792.20 814.68 836.55 857.85 878.64 898.94 918.78 938.21 957.23 975.87 994.17 1012.12 1029.76 1047.10
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.26. (continued) T (K)
p (mol/dm^)
0) (m/s)
3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.396 0.384 0.372 0.361 0.350 0.340 0.331 0.322 0.314 0.306 0.298 0.291 0.284 0.277 0.271 0.265 0.260 0.254 0.249 0.244 0.239
1064.15 1080.92 1097.44 1113.71 1129.74 1145.54 1161.12 1176.49 1191.66 1206.63 1221.42 1236.03 1250.46 1264.72 1278.82 1292.76 1306.55 1320.19 1333.68 1347.04 1360.25
TABLE 7.27. Air—Isobar at 50 MPa T (K)
p (mol/dm^)
0) (m/s)
100 200 300 400 500 600 700 800 900
30.94 21.58 15.23 11.68 9.54 8.10 7.06 6.28 5.66 5.15 4.74 4.38 4.08 3.82 3.59 3.38 3.20 3.04 2.89 2.76 2.64 2.53 2.43 2.33 2.24 2.16 2.09 2.02
1012.41 670.59 581.66 582.86 604.06 629.50 655.45 680.98 705.86 730.05 753.58 776.48 798.80 820.55 841.78 862.52 882.80 902.63 922.05 941.07 959.73 978.04 996.01 1013.67 1031.02 1048.10 1064.90 1081.44
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
{continues)
303
304
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.27.
(continued)
T (K)
p (mol/dm^)
2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
1.95 1.89 1.83 1.78 1.73 1.68 1.64 1.59 1.55 1.51 1.48 1.44 1.41 1.37 1.34 1.32 1.29 1.26 1.23 1.21 1.19 1.16
CO (m/s)
1097.73 1113.79 1129.62 1145.23 1160.63 1175.83 1190.84 1205.66 1220.30 1234.77 1249.06 1263.20 1277.18 1291.01 1304.69 1318.22 1331.62 1344.88 1358.01 1371.01 1383.88 1396.64
T A B L E 7.28.
Air—Isobar at 100 M P a
T (K)
p (mol/dm-^)
0) (m/s)
100 200 300 400 500 600 700 800 900
33.16 26.18 21.14 17.60 15.09 13.23 11.80 10.67 9.75 8.98 8.33 7.77 7.29 6.86 6.48 6.15 5.85 5.57 5.32 5.10 4.89 4.70 4.52 4.36 4.21 4.06 3.93 3.81
1192.37 932.01 818.47 779.80 772.41 778.24 790.14 805.13 821.78 839.35 857.39 875.65 893.95 912.18 930.28 948.19 965.90 983.39 1000.64 1017.65 1034.43 1050.98 1067.31 1083.41 1099.29 1114.97 1130.44 1145.72
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
REFERENCES
305
TABLE 7.28. (continued) T (K)
p (mol/dm^)
0) (m/s)
2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
3.69 3.58 3.48 3.38 3.29 3.20 3.12 3.04 2.97 2.90 2.83 2.76 2.70 2.64 2.59 2.53 2.48 2.43 2.38 2.34 2.29 2.25
1160.81 1175.72 1190.45 1205.01 1219.40 1233.63 1247.71 1261.63 1275.41 1289.04 1302.54 1315.90 1329.13 1342.24 1355.21 1368.07 1380.81 1393.43 1405.95 1418.35 1430.64 1442.83
Additional Reading Kinsler, L.E., Frey, A.R., Coppens, A.B., and Sanders, J.V. (1982). Fundamentals of Acoustics New York: John Wiley & Sons. Thompson, P.A. Compressible-Fluid Dynamics New York: McGraw Hill. (1972). Trusler, J.P.M. (1991). Physical Acoustics and Metrology of Fluids Bristol: Adam Hilger. Van Dael, W. (1975). In Experimental Thermodynamics of Non-Reacting Systems (International Union of Pure and Applied Chemistry, Commission on Thermodynamics and Thermochemistry; Le Neindre, B. and Vodar, B. eds. London: Butterworths. chapter 7, Thermodynamic Properties and the Velocity of Sound, pp. 527-575.
References 1. Slattery, J.C. (1981). Momentum, Energy, and Mass Transfer in Continua, New York: Krieger Pubhshing. 2. Thompson, P.A. (1972). Compressible-Fluid Dynamics New York: McGraw Hill. 3. NIST Thermophysical Properties of Pure Fluids Database (NIST12, Version 5.0). (2000). Natl. Inst. Stand. Tech., Gaithersburg, MD. 4. McQuarrie, D.A. (1976). Statistical Mechanics New York: Harper & Row.
306
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
5. Costa Gomes, M.F., and Trusler, J.P.M. (1998). The speed of sound in nitrogen at temperatures between T = 250 K and T = 350 K and at pressures up to 30 MPa, J. Chem. Thermodynamics 30: 527. 6. Trusler J.P.M. (1991). Physical Acoustics and Metrology of Fluids Bristol: Adam Hilger. 7. Van Dael, W. In Experimental Thermodynamics of Non-Reacting Systems (International Union of Pure and Applied Chemistry, Commission on Thermodynamics and Thermochemistry Le Neindre, B. and Vodar, B, eds. London: Butterworths, chapter 11: Thermodynamic properties and the velocity of sound, pp. 527-575. 8. Reid, R.C., Prausnitz, J.M., and Poling, B.E. (1987). The Properties of Gases and Liquids New York: McGraw Hill. 9. Camahan, N.F., and Starling, K.E. (1969). Equation of state for nonattracting rigid spheres, J. Chem. Phys. 51: 635. 10. Younglove, B.A. (1982). Thermophysical properties of fluids. I. argon, ethylene, parahydrogen, nitrogen, nitrogen trifluoride, and oxygen, J. Phys. Chem. Ref Data 11, pp. 1-1 to 1-353 (368 pages). 11. Sengers, J.V., and Levelt Sengers, J.M.H. (1978). In Progress in Liquid Physics (Croxton, C.A., ed., Chichester, U.K: Wiley. Chapter 4: Critical phenomena in classical fluids, pp. 103-174. 12. Tegeler, Ch., Span, R., and Wagner, W. (1999). A new equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressures up to 1000 MPa, J. Phys. Chem. Ref Data, 28: 779. 13. Estrada-Alexanders, A.F., and Trusler, J.P.M. (1995). The speed of sound in gaseous argon at temperatures between 110 K and 450 K and at pressures up to 19 MPa. J. Chem. Thermodyn. 27: 1075. 14. Span, R., Lemmon, E.W., Jacobsen, R.T, and Wagner, W. (1998). A reference quality equation of state for nitrogen. Int. J. Thermophys. 19: 1121. 15. lAWPS. (1995). Release on the Formulation for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use; copies of this and other lAPWS releases can be obtained from the Executive Secretary of lAPWS, Dr. R.B. Dooley, Electric Power Research Institute, 3412 Hill view Ave., Palo Alto, CA 94304 or at http://www.iapws.org. 16. Lemmon, E.W., Jacobsen, R.T, Penoncello, S.G., and Friend, D.G. (2000). Thermodynamic properties of air and mixtures of nitrogen, argon, and oxygen from 60 to 2000 K at pressures to 2000 MPa. J. Phys. Chem. Ref Data 29: 331. 17. Younglove, B.A., Frederick, N.V., and McCarty, R.D. (1993). Speed of Sound Data and Related Models for Mixtures of Natural Gas Components, NIST Monograph 178. 18. Lemmon, E.W., and Jacobsen, R.T. (1999). A Generalized Model for the Themodynamic Properties of Mixtures, Int. J. Thermophys. 20: 825. 19. Thermodynamics Research Center; Texas Engineering Experiment Station, College Station, TX (1961-2000): Natl. Inst. Stand. Tech., Boulder, CO (2000-).
Abstract In this Chapter, we review the principles of sound propagation in fluid systems. From a study of the hydrodynamic equations, sound propagation is shown to be a wave phenomenon. The speed of sound then can be derived at any state point from a knowledge of the thermodynamic surface of the fluid of interest. Several model equations of state are reviewed, and it is shown how the speed of sound can be obtained for a variety of systems. We then focus on several fluids of particular interest, and show the behavior of the sound speed over a wide range of the temperature and pressure variables. Tabulated values of the speed of sound are given for argon, nitrogen, water, and air based on the current standard reference thermodynamic surfaces.
7.1 Introduction In this Chapter, we discuss the propagation of sound in fluids and provide information about the thermodynamic speed of sound over substantial ranges of the state variables for a variety of fluids. In the context of this chapter, we consider sound to arise from a small periodic and isentropic (constant entropy) perturbation of the local equilibrium in a fluid, which, as we shall see, gives rise to a standard wave equation. The systems under consideration include both pure fluids and mixtures in the liquid, vapor, and supercritical states. Thus the range in temperature is from the melting line to very high temperatures (a dissociation limit), and the range in pressure is from very low values (below which the continuum approximation would not be valid) to the solidification locus (at least in principle). The major theme of the discussion stems from the relation 2-1
W = <
\dpdTj
M \
dp \
dp^
(7.1)
- p-
ar2
p
1/2
-J
237 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Vol. 39 ISBN 0-12-475986-6
Copyright © 2001 by Academic Press All rights of reproduction in any form reserved. ISSN 1079-4042/01 $35.00
238
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
which expresses the speed of sound, w, in terms of derivatives of the molar Helmholtz free energy A and the state point defined by the temperature T and the molar density p. In Eq. 7.1, M is the molar mass of the substance in kilograms per mole. Although Eq. 7.1 may not be the simplest expression for the speed of sound, it serves to emphasize that this thermodynamic quantity is easily calculable if the Helmholtz free energy of a fluid system is known. This expression is also appropriate for fluid mixtures, in which case the partial derivatives are to be taken at constant composition. In the next section, we outline how Eq. 7.1 can be obtained. Although this thermodynamic relationship is straightforward and exact, a few caveats must be given regarding its use. First "thermodynamic" sound model requires isentropic and linear approximations, and thus there are restrictions on the amplitude and frequency of the propagated disturbance. These issues are seldom significant when considering sound in most fluid media under most conditions. The second caveat is more essential: there are only a few fluid systems for which the Helmholtz free energy surface is sufficiently well known to allow sound speeds to be accurately calculated. These surfaces are generally empirical correlations of experimental data, and only when accurate sound speeds are included in the primary data used to determine the thermodynamic surface are the resultant calculated sound speeds obtained with small uncertainty. Experimental determinations of sound speeds, using state-of-theart techniques as discussed in other chapters, usually produce more accurate values for this quantity. However, more theoretical or predictive models for the Helmholtz free energy surface may be used to obtain useful approximations for the speed of sound for systems that have not been thoroughly characterized through experimental measurements. Once we have completed a derivation of Eq. 7.1 in the next section, we will explore some of the limits of this approach, and discuss the application to various model systems and to real fluids. Table 7.1 included at the end of this chapter provides reliable speed of sound values for a variety of important fluid systems.
7.2 Wave Equation in a Fluid From a standard understanding of thermodynamic conventions, it may be difficult to see how sound propagation can be considered a thermodynamic property. Despite a straightforward parsing of the name "thermodynamics," typical thermodynamic studies deal with unperturbed systems. Other responses to perturbing stresses, such as flow associated with shear stresses, heat flux related to thermal gradients, and diffusive flows caused by composition variations, are not considered thermodynamic in origin, and the separate study of steady-state kinetic theory may be required to obtain viscosity, thermal
WAVE EQUATION IN A FLUID
239
conductivity, and diffusion coefficients. The measurement of thermodynamic properties, as with most other physical quantities, may require a perturbing probe (i.e., a temperature change to determine heat capacities), but, in general, thermodynamic processes are equilibrium phenomena. As we will see, it is the linear, reversible, and small amplitude nature of the wave process that serves to connect sound propagation with fluid thermodynamics. 7.2.1 Hydrodynamic Equations When fluids are considered in the continuum approximation, three conservation laws can be used to determine the response to perturbing influences in an otherwise equilibrated system. These hydrodynamic equations enforce the conservation of mass (or the number of molecules of each species present), the conservation of momentum, and the conservation of energy. The continuity equation, ^
+ V . (pu) = 0
(7.2)
ensures that the molar density (at a given fixed infinitesimal volume in space) can change only with a flow of molecules into or out of the volume; here, the partial derivative is with respect to time t, and u represents the macroscopic velocity of fluid flow at the given point. We use the Euler convention in which the frame of reference is fixed in space, so that properties such as p are functions of position (r) and time, and they represent microscopic averages within a volume element centered on r in this fixed reference frame. The Navier-Stokes equation (7.3) f I - + u • v ) u = - - T ^ V(P - r;^ V • u) + - ^ V^u \dt J m Mp Mp is equivalent to Newton's second law and balances changes in momentum with forces acting on the fluid element. The operator ( ^ + u • V) is the material derivative or streaming derivative, and the left side of the equation describes changes in momentum due directly to fluid flow. The forces that act to change the momentum include the external force F, here divided by the mass m contained in the infinitesimal volume element under consideration, the force on the element due to gradients of pressure P, and forces associated with the viscous nature of any fluid: here, rjs is the bulk viscosity and rj is the usual shear viscosity. The pressure tensor has been decomposed, and a linear relationship between the stress and shear has been assumed [1]. Finally, the conservation of energy can be written in the form / d
\
R
X
^
240
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
where T is the absolute temperature, R is the molar gas constant, C^ is the molar isochoric heat capacity, and X is the coefficient of thermal conductivity in a linear, Fourier's law, relationship between thermal gradients and heat flux. Equations 7.2 to 7.4 are often simplified in the basic development of a wave equation formalism. In particular, we first consider an ideal fluid with no viscous effects or thermal conduction. In the absence of external forces, the Navier-Stokes relationship given by Eq. 7.3 reduces for an ideal fluid to the Euler equation.
Small disturbances of an equilibrium state can generate sound, and in the derivation of the wave equation, second-order effects in the derivatives can be ignored. Because the equilibrium state corresponds to zero macroscopic velocity, any quadratic terms in u will also be dropped. These approximations are almost always quite satisfactory in typical acoustic situations, and we will provide some quantitative discussion below. To lowest order, Eq. 7.5 can then be linearized and written as VP + M p , ^ = 0
(7.6)
where we have now introduced the notation XQ to denote the time-invariant and spatially invariant equilibrium value of a quantity. (In this notation, a physical quantity can be represented by x = x^ + X A , where XA is then the "acoustic" variable; i.e.. PA is the acoustic pressure, or the deviation of the pressure from its equilibrium value caused by a perturbation.) To the same order, Eq. 7.2 is written as ^
+ p„V • u = 0
(7.7)
01
Taking the time derivative of Eq. 7.7 and the divergence of Eq. 7.6 and subtracting the results gives M ^ - V 2 p = 0
(7.8)
Equation 7.8 forms the basis of the wave equation when we recall that the pressure and density can be related through the thermodynamic equation of state or Helmholtz energy surface. For a pure fluid, the pressure can be considered to be a function of the density and a second intensive thermodynamic variable (and for mixtures, the composition must also be given); without specifying the second variable, we write P = P(p,a), so that derivatives of the
WAVE EQUATION IN A FLUID
241
pressure can be written in terms of derivatives of the density and the parameter ot. Again we drop some higher-order terms and write Eq. 7.8 as M— 3/2
vV - V.
(7.9)
dp
If we can identify an appropriate variable a which remains constant through the perturbing acoustic signal (i.e., Va = 0), then Eq. 7.9 reduces to a standard propagating wave equation with wave speed [{\/M)dP/dp\a\^^^. The earliest theoretical acoustic derivations of wave propagation, developed by Newton, essentially assumed that the temperature is constant as the disturbance travels through the system. Although this may seem like a reasonable approximation, it is incorrect and leads to an incorrect value of the speed of sound. To lowest order, the situation can be further analyzed by examining the energy equation, Eq. 7.4, recalling that we are considering only the ideal fluid with A. = 0. We can then write + V ot
u
= 0
(7.10)
Ly
and use Eq. 7.7 to obtain a r
;^ r
R
'
/?
1
= 0 (7.11) where we have been consistent in keeping terms of lowest order and have assumed that the isochoric heat capacity is constant to this order. The time invariance of the quantity in the brackets means that Tp~^^^^ is constant during the perturbation. Using the ideal gas equation of state (discussed below), this leads to the constancy of Pp-(^v-\-R)/Cv^ which is a common expression for an adiabatic process in an ideal gas. (Recall that the isobaric heat capacity for an ideal gas is given by Cp = Cy -\- R, and the exponent of p is then y, the ratio of heat capacities, which has a value of 5/3 for an ideal monatomic gas [2].) Thus, at least in this limit, the perturbing force leads to an adiabatic process which implies that the entropy is constant during passage of an acoustic disturbance. This turns out generally to be a good approximation, and we choose entropy, S, for the variable a in Eq. 7.9. With V5 = 0, we can write the wave equation VV = 0
(7.12)
s to represent the hydrodynamic effect of a small disturbance on a fluid in equilibrium. In common with the interpretation of other wave equations in
242
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
physical systems, we conclude that the perturbation propagates unchanged with a propagation speed given by
^ = W T 7 ^
(7.13)
Equation 7.13 works remarkably well for a wide variety of fluid acoustic situations despite the approximations made in its derivation. This common expression for speed of sound is often not used directly, because derivatives at constant entropy are not often tabulated. Using a standard definition of the adiabatic compressibility, KS = {l/p)dp/dP\s, we can write w^ =
(7.14) MpKs
while additional standard thermodynamic relations allow us to write w' =
Cp
dP
MCy
dp
(7.15)
The relations derived from the Helmholtz free energy then give us the expression in Eq. 7.1. We conclude that a small-amplitude disturbance in an equilibrium ideal fluid will propagate as an isentropic wave with wave speed given by Eq. 7.1; this is a sound wave. 7.2.2 The Speed of Sound In statistical mechanics, an ideal gas is often considered to be a system of noninteracting point particles; the monatomic noble gases at low densities approximate this ideal condition. In this case, the equation of state can be written as P = p/^T, the molar isochoric heat capacity is generated from the three translational degrees of freedom as 3JR/2, and the isobaric heat capacity is 5RI2. For this system, Eq. 7.15 immediately gives a value of 322.59 m/s for the speed of sound in argon (M = 0.039 948 kg/mol) at 300 K. A currently accepted reference value at atmospheric pressure is 322.67 m/s, from which we can conclude that this simple theory and model works remarkably well. (Note that this degree of agreement between the properties of a real fluid and the ideal gas model should not generally be expected even at 1 atm, especially for heavier molecules.) Returning to our basic Eq. 7.1, note that the Helmholtz free energy for such a monatomic ideal gas can be written as A^>(p, T) = RT[A^-^\np
- 3/2 In J ]
(7.16)
WAVE EQUATION IN A FLUID
243
where A^ is an integration constant witli which we need not be concerned [4]. With this equation for A, the derivatives required for Eq. 7.1 are easily calculable with the result w' =
^-^ (7.17) 3M as previously seen for an ideal monatomic fluid. More generally, an ideal gas can have additional (intramolecular) degrees of freedom and internal structure. This is reflected in different values or temperature dependences in the ideal gas heat capacities of structured molecules, or, equivalently, in additional temperature dependence in the expression for the ideal gas Helmholtz free energy. With estimates for the translational, rotational, vibrational, and electronic energies based on spectroscopic information, the ideal gas heat capacities, using models as the rigid-rotor harmonic oscillator, can be calculated from statistical mechanics. From this information, values for the speed of sound of low-density systems can be calculated. Alternatively, accurate measurements of the speed of sound at low density can be used to provide estimates of the ideal gas properties of a molecular system. (However, if the frequency associated with the sound is not commensurate with the relaxation time of the modes contributing to the ideal gas calculations, corrections must be considered when connecting measured sound speeds to ideal gas behavior [5].) Interactions between the particles in a fluid (intermolecular effects) are extremely important when considering the speed of sound, or any other thermodynamic properties, outside the limited region in which the ideal gas approximation is vaUd. In this case, Eq. 7.16 is not an appropriate representation of the fluid's Helmholtz free energy, and Eq. 7.17 cannot be used to determine the speed of sound. Equations 7.1 and 7.13 through 7.15 are appropriate for compressed gases at elevated pressures, for saturated or compressed liquids, and for supercritical fluids. Explicit examples for the use of these equations in determining the speed of sound in a variety of fluids and over a range of state variables are given Section 7.4. Two general classes of approximations have been made in our derivation of the wave equation and the equivalent speed of sound expressions of Eqs. 7.1 and 7.13 through 7.15. These require that (1) the ampUtude of the wave be small so that the equations can be linearized, and (2) reversibility must be maintained so that isentropic behavior can be assumed. We shall briefly explore these requirements in this Section. The small-amplitude condition is easily fulfilled in most common examples of sound propagation. The restriction can be made that the maximum relative deviation of the density, the acoustic "condensation," is very small, \P - Po\/Po < < 1; alternatively, one can require that \P — Po\ << MpoW^. This condition can be satisfied if the pressure perturbations are much smaller than the equilibrium value of the pressure, which is typically the case.
244
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
To further quantify the small-ampHtude restriction, we note that the soundintensity level can be defined by a logarithmic ratio of an average energy flux in a propagating wave relative to a reference value for such a flux; SIL = 10 log(///ref), where SIL is the sound-intensity level expressed in decibels referenced to /ref. (Typically the reference value is 10~^^ W/m^ for sound in air, although common practices for this arbitrary reference depend on the application; e.g., for certain underwater acoustics applications, 6.76 x 10~^^ W/m^, equivalent to an acoustic pressure of about 1 jiPa, may be used.) The energy flux for a plane wave can be approximated as W^M(PA — Po)'^/(2po), where PA is the maximum value (or amplitude) of the density in the wave. In air at ambient conditions, the small-amplitude approximation breaks down (i.e., \P — Po\/Po = 1) at a sound-intensity level (re 10~^^ W/m^) of about 194 decibels (dB); for comparison, the range of hearing at 1000 Hz is about 0 to 140 dB — so the small-amplitude restriction holds for sounds with an energy of some 250,000 times that which may damage the human ear. For liquid water at ambient conditions, this condition would hold up to about 242 dB (re 10"^^ W/m^); with the reference based on 1 |iPa, this would be about 304 dB. We can conclude that, except in the cases of extraordinarily large "shock" pressures, the linear approximations hold for sound propagation. The more significant restriction on our derivation of the sound equation is associated with the irreversibility caused by the viscous and heat flux effects that were ignored despite their importance in Eqs. 7.3 and 7.4. Additionally, the inability of a fluid to instantaneously return to an equilibrium state when perturbed can be related to the relaxation times that govern the fluid's behavior. The energy in a molecular system is distributed among internal (such as vibrational and rotational modes) and translational degrees of freedom; when the time scales for populating these energy levels are not compatible with the frequency of the propagating sound, the derivations given previously will not be accurate. We will not go into details here, but these effects lead to both attenuation of sound and dispersion — the dependence of the speed of sound on the frequency. One can make various approximations to the full hydrodynamic equations to obtain corrections to the speed of sound equations given previously. The equations can be solved in the complex plane, and to lowest order, the viscous and thermal effects are found to leave the speed of sound unchanged, and contribute only to a frequency-dependent absorption [6]. More generally, one can obtain an expansion of the form w^(f) = w^{0)[l + aX + 0(X^)]
(7.18)
for the real part of the speed of sound as a function of the frequency, / ; the expansion parameter, X, is related to the relaxation times and the frequency.
EQUATIONS OF STATE FOR FLUIDS
245
and the coefficient a can be evaluated for certain models or treated as an empirical parameter. The solution given by Eq. 7.1 represents the zero frequency limit, and, depending on the particular relaxation mechanisms present, some dispersion effects may be seen at frequencies above 1 GHz. The frequency at which dispersion is important for a given system depends on the rotational and vibrational relaxation times in the molecule; these are generally related to the molecular symmetry. Note that the high-frequency signals correspond to small wavelengths, and it is at small wavelengths that the continuum approximation, essential in the development of hydrodynamic approaches to the fluid state, is inappropriate. Finally, we note that the fluctuation-driven phenomena and the density correlations present in near-critical fluids will have an effect on the isentropic assumption and may contribute to dispersion at even lower frequencies. Details concerning these aspects of the acoustic approximation can be found in the texts listed in "Additional Reading."
7.3 Equations of State for Fluids From the previous discussion, we conclude that the speed of sound in a fluid can be determined from its Helmholtz free-energy surface; see Eq. 7.1. In general, an expression for A is formed from the sum of ideal gas and residual contributions: A{pT) = A'^(p, T) + A'*(p, T). The ideal-gas term represents energy associated with noninteracting particles and their internal degrees of freedom—i.e., from the statistical partition function involved with translational, electronic, vibrational, and rotational motions. This quantity can be computed from models of the molecular architecture (i.e., rigid-rotator, harmonic oscillator model for nonlinear polyatomic molecules) and spectroscopic data. Following the format of Eq. 7.16, the ideal gas portion of the Helmholtz energy can be written in general as A^HP, T) = RT[A^ + Inp + f(T)]
(7.19)
where / gives all of the temperature dependence. It is often sufficient to consider only the A^^ contribution for gas phase systems at very low density; however, the residual contribution becomes important as the pressure is increased. The residual contribution to A describes the effects of interparticle interactions. An equation of state, typically an expression for the pressure as a function of temperature and density (or volume), can also be decomposed into the ideal gas and residual contributions. The ideal-gas term in the equation of state is known theoretically as P^^ = pRT, and it is the ideal gas heat capacity (or related thermodynamic quantity) that is used to obtain A'^. Accurate expressions for the residual contributions to A or P are generally based on empirical
246
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
thermodynamic data. The same decomposition of the Helmholtz free energy into ideal and residual parts holds also for mixtures; in this case, the ideal gas contribution is simply related to a mole fraction average of the pure components' ideal gas values, but cross-interaction effects make the residual values more difficult to determine. In Section 7.4, we see how these two parts of A contribute to the speed of sound in fluids. In Figures 7.1 and 7.2 we sketch the basic phase topology of simple fluid systems to understand the scope of the Helmholtz energy formulation for fluid systems. All pure fluids have phase diagrams similar to Figure 7.1; however, quantum fluids may look somewhat different at low temperatures, and some fluids may decompose at temperatures within the area depicted. The solid-fluid phase boundary intersects the vapor-liquid equilibrium (VLB) phase boundary at the triple point, a single value of temperature and pressure at which the solid, liquid, and vapor phases coexist. The melting line extends, in principle, to infinite pressures (although there are, of course, other phenomena at extreme conditions), but the VLB phase boundary terminates at a critical point. For temperatures above the critical-point temperature, the fluid is said to be in the supercritical phase; for lower temperatures, the fluid is a vapor at pressures below the VLB phase boundary and a liquid at higher pressures (to the solidfluid boundary). Along the VLB phase boundary, the fluid is said to be in the saturated liquid or saturated vapor state; inside the two-phase region of
Critical point
Supercritical fluid
Temperature
FIG. 7.1. Pressure vs temperature phase diagram of a simple pure fluid. SLE is solidliquid equilibrium and VLB is vapor-liquid equilibrium.
24'
EQUATIONS OF STATE FOR FLUIDS
Density FIG. 7.2. Pressure vs density phase diagram for a simple pure fluid. Figure 7.2, the equilibrium state consists of two phases, and the speed of sound is not uniquely defined. The phase topology for mixtures can become significantly more complex. In general, for a system at fixed composition, the VLE phase boundary is no longer a simple line. The liquid will vaporize at a bubble point, and the bubbles may have a composition different from that of the coexisting liquid. The vapor condenses at a dew point, and the coexisting liquid may also have a different composition. The locus of dew points and bubble points for a fixed composition may be joined by a critical point. In addition, there may be liquid-liquid immiscibility in portions of the phase diagram. The speed of sound is well defined for each one-phase point, and the same equations given previously can be used for mixtures. 7.3.1 Simple Equations of State There are several ways to represent the residual portion of the thermodynamic surfaces of fluids. The virial series is a density expansion for low densities based on statistical-mechanics cluster approaches [4]. The pressure can be written as P = pRT[\ + B{T)p + C{T)p^ + . . . = pRT[\ + ^
Bi{T)p']
(7.20)
In this notation, B is the second virial coefficient, but Bi is the (/ + 1) virial coefficient. Although the second and, perhaps, third virial coefficients can
248
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
be determined from experimental data, virial coefficients of higher order are difficult to measure, and the series itself does not converge as the critical density is approached. Equation 7.20 is rigorous in the low density limit, and for gases at low and moderate density, up to about one third of the critical density, this type of representation can be quite accurate. Equation 7.20 yields (7.21) and, using Eq. 7.1, the speed of sound for the virial equation of state is given by
w^ =
M
RT + RTY^{i + \)Bip' (=1
R+2RTY.^,(^
+ B}jp^ + RT^Y.i,i=x{^ + B}j{^ + B^p'+i
(7.22) where primes and double primes indicate the first and second temperature derivatives, respectively, of the virial coefficients, Bi and the temperature term in A'^, / from Eq. 7.19. It is straightforward but tedious to expand the denominator in Eq. 7.22 to write a density series of the form 2 ^ry" w = M
l + ^/^Kr)p'
(7.23)
1=1
where y" is the ratio of the ideal gas heat capacities, 1 — ( 2 7 / + T^/") in the current notation. The first two terms are given by [7] '
KxiJ) = 2Bi + liy" - l)TBi +
(1 - y " ) ^
T "
^T^B^
(7.24)
and y" - 1 K2iT) = ^—^[Bi yo + ^ yo
, T „ -, 2y'' + 1 + ily" - \)TB, - (1 - Y'')T^B,f + - ^yO— B 2 TB. + ^
-T^B. 2y°
(7.25)
\
EQUATIONS OF STATE FOR FLUIDS
249
Thus, if the virial coefficients are known, they can be used to calculate the speed of sound in a gas. In these expressions, we see the intimate connection between the ideal gas value of the speed of sound and the full expression for this property. In Eq. 7.23, the ideal gas contribution is identified with the zero-density value, w^^ = (RTy^/My^^, but note that a contribution from internal degrees of freedom, y^, also affects the higher order terms in the expression for the speed of sound as seen in Eqs. 7.24 and 7.25. It is generally true that the ideal gas speed of sound is consistent with the low-density limit of a thermodynamic expression, and further that the internal modes in a molecular system also affect the sound speed at high densities. The family of cubic equations of state represents another simple form to describe the thermodynamic surface. In general these equations can be written as [8] pRT ap^ (7.26) P = \ —bp \-\- ubp -\- vb^p^ where u and w serve to distinguish between the type of cubic equation (e.g., van der Waals, Peng-Robinson, etc.), b is typically a fluid dependent constant, and a is also fluid dependent, but may be constant or a function of temperature, depending on the model being considered. The residual Helmholtz free energy can then be written as A'(p,T)
=
-RTln(l-bp)
2-j-bp(u-\-^/u^ -4v) In p\lu^ — Av 2-\-bp{u — yfu^ Av) (7.27)
and the speed of sound becomes w
ap(2-\- bup) [vb^p^ + bup + \f
RT M L (l-bp)^
\_R L1 - ^P 2Rf
-\- RTf"
a'p 1' vb'^p'^ + bup-{-\\ 2-hbp(u-^y/u^ -4v) In b^u^ - 4u ' " 2 -h bp{u H- y/u^ - Av)
(7.28)
The primes again indicate temperature derivatives. The expression simplifies considerably for the original van der Waals equation (where u = v = 0 and a is constant), but the general expression of Eq. 7.28 can be used for any of the common cubic equations of state. (Note that in the case of w = w; = 0, the logarithmic expression can be expanded, and the full equation is still well defined.)
250
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Cubic equations are often used to describe fluids in engineering applications. These equations can provide reasonably good descriptions of a thermodynamic surface, (except near the critical point), and liquid-vapor phase boundary information can be calculated directly from this approach. The sound speeds calculated from expressions such as that given previously may be adequate approximations. For more accurate descriptions of the speed of sound (and other properties), the reference quality equations, as discussed in Section 7.3.2, are required. In addition to these simple equations that can be used to approximate the properties of real fluid systems, we mention that there are also model systems for which properties can be calculated. For instance, the hard-sphere fluid, consisting of (structureless) spherical particles of diameter a and relative mass M, interacts with a potential function which is infinite when the particles collide and zero otherwise. This model forms the basis for many computer simulations and theoretical results and is often applied to real systems. The equation of state for this fluid has been shown empirically to be well described by the Camahan-Starling expression [9] P = pRT
^ ; / \3 (1 - r]y
'
(7.29)
where rj = 7ta^pNA/6 and NA is the Avogadro number. This leads to a Helmholtz energy expression of the form A^/^rJA^+lnp-^lnr + l ^ ^ ^ j
(7.30)
and Eq. 7.1 immediately gives 2
RT \5 + lOr]- 3rf- - 24rj^ + 31rr' - 22r]^ + 5??^ 1
" = 3M 1
(T^^
)
^'-''^
Unfortunately, we are not aware of any computer simulations which have generated the speed of sound of the hard-sphere system, but the model can be used for real fluids with an effective hard-sphere diameter regressed from available data. Of course, alternative ideal gas behavior can also be incorporated into such a hard-sphere model for real fluids. 7.3.2 Other Equation of State Descriptions The most accurate equations of state are those that seek to represent all available thermodynamic data to within their experimental uncertainty. These are generally expressed by the pressure as a function of temperature and density together with a correlation for the ideal gas properties, or directly
EQUATIONS OF STATE FOR FLUIDS
251
as the Helmholtz free energy as a function of temperature and density. Such equations are available for industrially important fluids and are suitable for purposes such as instrument calibration and custody transfer applications. These equations are developed by regression of large numbers of data over a broad range of state variables and may have 30 to 40 adjustable coefficients to describe the vapor, liquid, and supercritical phases, including the phase boundary information. Of course there are also thermodynamic surfaces of intermediate complexity compared with the simple cubic or virial coefficient representations and the reference quality equations being considered here. Examples of the accurate reference quality equations of state include the Benedict-Webb-Rubin family of equations, which have the form of P = pRT + J^ Gip'^^T'^^ + Yl GiP'^'T'^' exp(}/p2) i=\
(7.32)
i=Ni
Typical equations of this form [10] have 32 coefficients G/, with the exponents of density ranging from 2 to 13 and the exponents of temperature ranging from —4 to ^; several important fluids have been described by this functional form with fixed exponents, nt and m,, by regressing only the coefficients Gt based on the available experimental data on density, heat capacity, phase boundary, speed of sound, etc. This type of equation can also be converted into an expression for the Helmholtz free energy, and the speed of sound can be calculated from Eq. 7.1 as before. Recently, several equations of the form A(p, T) = A'^{p, r ) + ^
G.-^'r'^' + 2
i=\
G,(5"'T'"'- exp((5^0
i=Ni
N
+ X]G,^"'^'"'exp[g(5,r)]
(7.33)
i=N2
have been developed for important fluids, where 8 = pjPc, t = r ^ / r , the subscript "c" indicates a value at the critical point, and ^ is a function that improves the description of the thermodynamic surface in the critical region. These equations are also determined empirically from the best available experimental data; in this case, the numbers of terms in each sum, the exponents «/, m/, //, and the function g may be optimized from the data. The resultant "structurally optimized" thermodynamic surface generally provides the best general representation over the full range of fluid states, and the speed of sound determined using Eqs. 7.1 and 7.33 provide an accurate estimate of the property. The sounds speed tabulated in Section 7.6 were generated from models such as those of Eqs. 7.32 and 7.33.
252
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
When reference-quality thermodynamic surfaces are not available for a fluid of interest, predictive models for the surface can be used to determine the speed of sound. The residual Helmholtz energy surfaces of similar fluids generally obey a corresponding states principle [4, 8]; thus, with two-scale factors, representing changes in energy and distance scales, one can generate an approximate surface for a fluid based on the thermodynamic surface of a more well-studied fluid. This approach allows reasonable estimates of the speed of sound with a minimal amount of information. The cubic equations of state of Eq. 7.26 can be implemented when values for the critical temperature and pressure are known; the more complex cubic equations require additional information, and other extensions of the corresponding states principle can be implemented that make use of available experimental information. The molecular structure must be considered explicitly to approximate the ideal-gas effects, which are quite significant; group additivity and related methods can be useful for this problem when spectroscopic data are not available. Direct predictions of the heat capacities and compressibilities can also be made; these enter into the speed of sound calculation using Eq. 7.15. Alternatively, corresponding states or related models can be developed for the speed of sound itself, and these can be considered independently of the full Helmholtz energy surface. Fluid mixtures generally obey the same thermodynamic and hydrodynamic principles as pure fluids, and thus equations of state can be constructed that look similar to those discussed previously, and the speed of sound can be computed using the same equations, with derivatives being taken at fixed composition. Among the special considerations, we note that the continuity equation, Eq. 7.2, must be revised to reflect conservation of each species present, and the irreversibilities associated with demixing upon passage of a sound perturbation must be considered at high frequencies. In addition, as mentioned previously, the phase topology of mixtures generally differs from that of pure fluids, and care must be taken with any calculation of the speed of sound that the appropriate phase has been identified. Thermodynamic surfaces for mixtures can be calculated from a virial equation of state if the pure fluid coefficients are known and the relevant "cross" virial coefficients, representing interactions between the distinct species in the fluid mixture, can be estimated. Cubic equations of state can also be extended to mixtures using well-defined mixing rules and, if available, a set of interaction parameters representing the cross-species effects. Reference quality equations can also be extended to mixtures with appropriate mixing rules; available experimental data over a range of compositions can be used to regress additional correction terms and hence improve the mixture calculations. As a simple, but explicit, example of a calculation of the speed of sound in a mixture, we consider a system at very low densities, so that the ideal
EQUATIONS OF STATE FOR FLUIDS
253
gas equations are appropriate. The equation of state, P = pRT, remains valid for ideal gas mixtures, and we denote the isobaric and isochoric ideal gas heat capacities of a mixture as C^p^^^ and Cy j^^^. Equation 7.15 can be expressed as
^y^'^f^^la^IL V
mix/
ig
yi^ .
(7.34) V
/
^V,mix ^^"^"^^^
Note again that this is the same as for a pure fluid, e.g. Eq. 7.17, although in that case, we considered a monatomic gas with heat capacities 3R/2 and 5R/2. The pure components of the mixture (denoted by a subscript /) more generally have ideal gas heat capacities C^- and C^ •, and they are present with a mole fraction of JC/. It is rigorously correct that the molar mass and ideal gas heat capacities of a mixture can be written as mole-fraction-weighted averages of these quantities for the pure components: Mmix = S-^/M; C^p,mix — ^^i^^p,i''> C(f jj^.^ = Ejc/Cif.. Then Eq. 7.34 can be written as
From Eq. 7.35 we see that the speed of sound of a mixture, even in an ideal gas state, cannot generally be calculated as a mole fraction average of the speed of sound of the components. Near the critical point of a pure fluid or a mixture, none of the equations of state discussed earlier is adequate. At this singularity in the thermodynamic surface, the behavior is governed by fluctuations that have very long length scales rather than by the comparatively short-ranged interactions observed in other regions of the phase diagram [11]. The phenomenon of "critical slowing" affects the hydrodynamics of a near-critical system, so the derivation of the speed of sound as a thermodynamic property must be reexamined [7]. The thermodynamic speed of sound is strictly zero at the critical point, which can be seen by Eq. 7.15 with the additional observation that the strong critical divergence of the isobaric heat capacity is compensated by a similar decrease in the isothermal derivative; thus, the weak divergence of the isochoric heat capacity in the denominator of the expression ensures that the speed of sound approaches zero at the critical point. This value of the speed of sound can be obtained from a classical Helmholtz free energy surface, although a correct description of the approach at the critical point requires a nonanalytical, scaling theory expression for the free energy that is outside the scope of this chapter. The dispersion of sound (i.e., the dependence of the speed of sound on frequency) is very pronounced in the critical region, and it is the low-frequency limit that is governed by these thermodynamic considerations.
254
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
In summary, the thermodynamic speed of sound of a fluid system at any temperature, density (or pressure), and composition can be obtained by using Eq. 7.1 if the Helmholtz free energy surface for the fluid is known. The quality of the speed of sound calculated in this manner of course depends on the quality of the thermodynamic surface that is used. The reference quality equations can reliably be used, as they should reflect the uncertainty of the available speed of sound data.
7.4 Speed Of Sound in Fluids Argon is one of the simplest fluid systems because of its location on the periodic table, and it is also an important commodity chemical. The system has been well studied, and the reference quality Helmholtz energy equation of Tegeler et al. [12] provides an excellent basis for examining the behavior of the speed of sound in argon. The equation of [12] is of the form described by Eq. 7.33; in addition, a modified Benedict-Webb-Rubin equation as in Eq. 7.32 is available in [10]. In Figures 7.3 to 7.5, we show the behavior of the speed of sound in various projections over a large range of the state variables. These and other figures are based on the sound speed calculations available in the NIST pure fluids database [3]. In Figure 7.3, the saturation boundary is shown in bold; at a Argon
80
380
680
Temperature, K
FIG. 7.3. The speed of sound in argon vs temperature along isobars. The heavy lines give values on the saturated liquid and saturated vapor curves.
255
SPEED OF SOUND IN FLUIDS
Argon 1000
30 40 Pressure, MPa FIG. 7.4. The speed of sound in argon vs pressure along isotherms. The heavy Hnes give values on the saturated liquid and saturated vapor curves.
Argon
185
235 285 Temperature, K
335
385
FIG. 7.5. The speed of sound in argon vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves.
256
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Argon
0.03 0.02 0.01
I
0.00
CO
!'#lM'iS
-0.01 H
'
I
I
250
300
350
tr'
-0.02 -I -0.03
100
150
200
400
450
Temperature, K
FIG. 7.6. Sample deviations between experimental and calculated values for the speed of sound in argon. Two points (with deviations of —0.26% and 0.54%) are not shown. fixed temperature, the two lines represent the sound speed in the coexisting vapor and liquid. Note that the sound speeds approach zero at the critical point. Note from Figures 7.3 and 7.4 that the speed of sound isotherms cross in these pressure projections. The isochores (lines of constant density) are also seen to cross in Figure 7.5; for supercritical fluids, the speed of sound tends to increase with density and with temperature along the isochores. In Figure 7.6, we compare the calculated sound speeds with the data of Estrada-Alexanders and Trusler [13]. The 173 data were obtain between 110 K and 450 K for pressures ranging from 6.83 kPa to 19.26 MPa using a spherical acoustic resonator with experimental uncertainties from 0.001 to 0.007%. These data were used in the regression of the thermodynamic surface, and the deviations between the data and this reference quality surface are consistent with the experimental uncertainty assessment. In general, as discussed previously the reference quality equations of state provide calculations of the speed of sound that are consistent with the experimental data. In Figure 7.7, we compare the 300 K sound speed isotherm for argon as calculated from the equation of state of [12] with the results of various models. A density of 30 mol dm~^ for this temperature corresponds to a pressure of about 190 MPa. The dashed line gives the ideal gas value for argon at 300 K, as calculated from Eq. 7.17. This quantity is independent of density, and agrees quite well with the reference equation of state at zero density. In fact, all of the models agree with Eq. 7.17 in this limit. We have included
257
SPEED OF SOUND IN FLUIDS
Argon: 300 K
Virial: 3PE feng Robinson
Virial: 2PE
0
5
10
15
20
25
30
35
Density, mol dm~^
FIG. 7.7. Speed of sound of argon vs density at 300 K. The bold line labeled EOS represents values calculated from the standard reference equation of state. The other lines are values calculated from various models discussed in the text. three lines to represent the virial expressions of Eqs. 7.20 to 7.25. The line labeled "Virial: 2E0S" used the first two acoustic virial coefficients [Eqs. 7.24 and 7.25], where the pressure virial coefficients and their derivatives were calculated numerically from a computer implementation of the reference thermodynamic surface of Tegeler et al. [12]. The line "Virial: 2PE" used the first two acoustic virial coefficients tabulated by Estrada-Alexanders and Trusler [13] that had been determined from an interatomic potential energy function. The differences in these two lines come about because of the difficulty in numerically determining the second temperature derivatives of the pressure virial coefficients and the sensitivity of the acoustic virial coefficients to these quantities. The model ("Virial: 3PE"), using three virial coefficients as tabulated by Estrada-Alexanders and Trusler [13], is seen to provide the best representation of the reference line even to the high pressures which are shown. The van der Waals and Peng-Robinson equations of state are two forms in the family of cubic equations of state. The parameters of both of these equations are determined from the critical parameters of argon and, in the case of the Peng-Robinson equation, the acentric factor for argon. For the van der Waals equations, both u and v of Eq. 7.28 are zero; for the Peng-Robinson equation of state w = 2 and u = — 1. These equations may have problems reproducing experimental densities at high values of the pressure, and the
258
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
speeds of sound are seen to deviate significantly from the reference line. Finally, Figure 7.7 shows the results for the speed of sound of the hard sphere fluid as given in Eq. 7.31. Only a value of the effective hard sphere diameter, a is needed in order to evaluated the parameter rj. We have determined a = 2.88 X 10~^^ m by requiring that the hard sphere result be consistent with the sound speed of argon at 20 mol dm~^. The hard sphere expression seems to give somewhat less curvature than the reference line for this isotherm. Nitrogen provides our second example, and in Figures 7.8 to 7.10 we indicate the behavior of the speed of sound based on the Helmholtz energy formulation of Span et al. [14]. The general behaviors of the curves shown are seen to be quit similar to those of argon; for similar state points, nitrogen generally has a larger value of the speed of sound. Figure 7.11 compares the calculated speed of sound with high-quality experimental measurements on argon of Costa Gomes and Trusler [5]. The data were obtained in a spherical resonator operating at frequencies between 5 kHz and 26 kHz, and the experimental uncertainties were estimated [5] as less than 0.01%. A difference between the measured results at these frequencies and the thermodynamic speed of sound, corresponding to zero frequency, was noted by Costa Gomes and Trusler. Figure 7.11 also shows the data as adjusted to zero frequency in [5]. These zero-frequency data were considered in the regression of Span
Nitrogen 2000-1
1800 J
500 MPa
1600-^ CO
E
1400 J
c 1200 H o w 1000H 'o "D 800 H 0
600-^ 400 200 0
T
1
1
1
1
1
T
110 160 210 260 310 360 410 460 510 560 Temperature, K
FIG. 7.8. The speed of sound in nitrogen vs temperature along isobars. The heavy lines give values on the saturated liquid and saturated vapor curves.
259
SPEED OF SOUND IN FLUIDS
Nitrogen 100 K
900 n
0 0
5" 300 H
I
10
20
T 1 r 30 40 50 Pressure, MPa
r 60
T 70
"I 80
FIG. 7.9. The speed of sound in nitrogen vs pressure along isotherms. The heavy Unes give values on the saturated liquid and saturated vapor curves.
Nitrogen lOOOn
225 275 Temperature, K
FIG. 7.10. The speed of sound in nitrogen vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves.
260
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Nitrogen
0.01 0.00 -0.01
m
* la i
+AA SAA
I I
^
i
-0.02 -0.03 -0.04
'•••-.
-0.05 10
15
20
25
30
Pressure, MPa
FIG. 7.11. Sample deviations between experimental and calculated values for the speed of sound in nitrogen. The closed symbols represent data obtained at frequencies between 5 kHz and 26 kHz; the open symbols are data corrected to zero frequency. Circles: 250 K; squares: 275 K; triangles: 300 K; diamonds: 350 K. et al. [14] and, as can be seen, the reference equation of state agrees with the data to within their uncertainty. The properties of water are well described by the standard formulation adopted by the International Association for the Properties of Water and Steam (lAPWS) [15]. Figures 7.12 to 7.14 show the behavior of the speed of sound of fluid water. Because of the relatively high critical temperature of water (647.096 K) relative to that of argon and nitrogen, the region of phase equilibria is expanded in these figures. The vertical line segments in Figures 7.12 and 7.13 connected the isobars and isotherms, respectively, across the two-phase region. This is the region of liquid-vapor phase separation, and the thermodynamic speed of sound is not defined for the two-phase systems. Figure 7.12 includes a line representing the critical isobar (Pc = 22.064 MPa); this isobar intersects the phase boundary at the critical temperature, whereas all subcritical isobars meet the saturation boundary at liquid and vapor points, and supercritical isobars do not intersect the phase boundary. As mentioned earlier the thermodynamic speed of sound should vanish at the critical point, however, the classical equation for water gives a small but non-zero value. For some equations of state, this value may depend on the computer implementation and the particular calculation performed: a value of about 35 m/s can be obtained on the phase boundary at the critical temperature for the lAPWS equation.
261
SPEED OF SOUND IN FLUIDS
Water
500 MPa
Temperature, K
FIG. 7.12. The speed of sound in water vs temperature along isobars. The heavy lines give values on the saturated liquid and saturated vapor curves. Water
1600-1
400 K
1400 HV
275 K
300 K 500 K
w 1200-^
1 1000 H o CO
° 800-^ ^
600 H
700 K _^
Nw
400-]
200-1
1
1
1
1
0
2
4
6
8 10 12 14 16 18 20 22 24
1
1
1
1
1
1
1—r~
Pressure, MPa
FIG. 7.13. The speed of sound in water vs pressure along isotherms. The heavy lines give values on the saturated liquid and saturated vapor curves.
262
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Several features of the behavior of the speed of sound in water can be observed in Figure 7.13. Note that there is a small region where the speed of sound in the saturated vapor (the lower branch of the phase boundary in most of the figures) is greater than that in the coexisting liquid. This occurs for temperatures between about 643 K and the critical temperature. Near the pure-fluid critical point, the vapor and liquid states become quite similar; although the vapor has a smaller density, the speed of sound in the vapor is larger than that of the liquid in this region. The subcritical isotherms in Figure 7.13 illustrate another feature: note that for fixed pressure (at about 18 MPa, to be specific), the speed of sound at 300 K and 400 K is significantly higher than that at 275 K, and the 500 K isotherm is inverted compared to the trend of the others. This behavior is related to the maxima seen in the isobars in Figure 7.12. Figure 7.14 shows the clear maximum of the speed of sound in the saturated liquid and the broader maximum in that saturated vapor phase. The behavior of the speed of sound in air is shown in Figures 7.15 to 7.17. These values were calculated from the thermodynamic surface of Lemmon et al. [16] which provides a reference quality equation of state for dry air considered at the fixed composition of 0.7812 nitrogen, 0.2096 oxygen, and 0.0092 argon by mole fraction. We see that the general behavior of the mixture isobars (Fig. 7.15), isotherms (Fig. 7.16), and isochores (Fig. 7.17) is quite similar to the trends seen for the pure fluids. Note that the subcritical isobar at 0.1 MPa also shows a segment in the two-phase region. This line is Water 1600
275
375
475
575
675
775
T 875
1175
Temperature, K
FIG. 7.14. The speed of sound in water vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves.
263
SPEED OF SOUND IN FLUIDS
Air 1150
950
•D C 13 O (0
750
550 •D
0 0 Q.
350 H
150 60
160
260
360 Temperature, K
460
560
FIG. 7.15. The speed of sound in air vs temperature along isobars. The heavy Unes give values on the saturated liquid and saturated vapor curves (bubble-point curve and dew-point curve).
Air
30 Pressure, MPa
FIG. 7.16. The speed of sound in air vs pressure along isotherms. The heavy hues give values on the saturated liquid and saturated vapor curves (bubble-point curve and dew-point curve).
264
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
Air »uu20 mol dm-3
700^
|600-
10 mol dm~3 5 mol dm'
§500o CO
O400•D 0) CD
(g-300-
0.1 mol
^ ~ \ ^ ^
200100-
60
1
1
11
,
,
,
,
110
160
210
260
310
360
410
T
460
1
1
510
560
Temperature
FIG. 7.17. The speed of sound in air vs temperature along isochores. The heavy lines give values on the saturated liquid and saturated vapor curves (bubble-point curve and dew-point curve). not vertical, however, and does not connect states in equilibrium. The lower saturation boundary is the line of dew points. At 0.1 MPa, the dew-point temperature for the composition specified is at about 81.6 K. The bubble-point temperature is at about 78.8 K. The bubbles in equilibrium with the bulk liquid and the condensate in equilibrium with the bulk vapor have different compositions than the bulk fluid. Again, the thermodynamic speed of sound is not defined in the two-phase region. Finally, in Figure 7.18 we show some deviations between experimental sound speeds for a natural gas fluid and calculations from a model for the mixture Helmholtz free energy. The data were obtained with a cylindrical cavity for a methane-rich ten-component mixture representing Gulf Coast natural gas [17]. The experimental uncertainty is about 0.05%. The generalized mixture model representing the base line [18] uses the reference quality equation of state for each pure component and a few mixture parameters based primarily on density and phase equilibrium data. The model is seen to agree with the data to nearly within the experimental uncertainty.
7.5 Conclusions The thermodynamic properties of a fluid system are completely determined by the Helmholtz free energy surface of the system. We have seen that the
265
CONCLUSIONS
Natural gas 0.14 0.12 -I
0.1 •] ^
0.08 \
c •B 0.06 CO
Q
•
0.04 A 0.02
• • • " •
•
— •
0 H -0.02 4
6 Pressure, MPa
8
10
12
FIG. 7.18. Sample deviations between experimental and calculated values for the speed of sound in a natural gas. speed of sound in such a system can be considered a thermodynamic property, and that it can be calculated from Eq. 7.1 The available sources for the speed of sound in fluid systems can be considered to be of three types: (1) experimental measurements and compilations of experimental measurements can provide the most accurate values for the speed of sound in vv^ell-defined systems; (2) reference quality Helmholtz free energy surfaces have been developed for important fluids and can describe the speed of sound essentially to within the uncertainty of available data; (3) models or predictive schemes can be used for systems for which measurements are not available, including more complex molecular systems and mixtures. Among the sources for experimental data, the tables [19] of the NIST Thermodynamics Research Center provide an extensive compilation for both hydrocarbon and non-hydrocarbon systems. These tables are available in both printed and electronic form. Researchers at the National Institute of Standards and Technology have developed several reference thermodynamic surfaces based on experimental measurements at NIST and elsewhere, and have compiled reference quality surfaces from several other sources. Such information is available in the form of PC-based computer programs such as the NIST Thermophysical Properties of Pure Fluids Database [3]. NIST also provides predictive computer packages that can be used for pure fluids with limited measurements and for mixtures. The catalog of computer packages from the Standard Reference Data
266
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
office of NIST is available on-line at http://www.mst.gov/srd. Some information on the thermophysical properties of fluids is available in the NIST chemistry webbook at http://webbook.nist.gov. The material in Tables 7.1 through 7.4 was calculated from these PC computer programs.
7.6 Tables of the Speed of Sound in Important Fluids TABLE 7.1. Argon — Saturation Boundary Vapor
Liquid
r(K) 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
P (MPa)
p (mol/dm-^)
CO (m/s)
p (mol/dm-^)
0) (m/s)
0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.15 0.16 0.18 0.19 0.21 0.23 0.25 0.28 0.30 0.32 0.35 0.38 0.41 0.44 0.47 0.51 0.54 0.58 0.62 0.67 0.71 0.76 0.81 0.86 0.91 0.97 1.02 1.08 1.15 1.21 1.28 1.35 1.43 1.50 1.58 1.66 1.75 1.84
35.44 35.28 35.13 34.98 34.82 34.67 34.51 34.35 34.19 34.04 33.87 33.71 33.55 33.39 33.22 33.05 32.89 32.72 32.54 32.37 32.20 32.02 31.84 31.66 31.48 31.30 31.11 30.92 30.73 30.54 30.34 30.14 29.94 29.74 29.53 29.32 29.11 28.89 28.67 28.45 28.22 27.98 27.75 27.50 27.25
861.10 854.24 847.35 840.43 833.47 826.48 819.45 812.39 805.28 798.14 790.95 783.72 776.45 769.14 761.78 754.37 746.91 739.40 731.83 724.21 716.54 708.81 701.02 693.16 685.24 677.26 669.20 661.08 652.87 644.60 636.24 627.79 619.26 610.64 601.93 593.11 584.19 575.16 566.02 556.75 547.36 537.83 528.16 518.34 508.36
0.104 0.115 0.127 0.140 0.155 0.170 0.186 0.204 0.222 0.242 0.264 0.286 0.310 0.336 0.363 0.392 0.422 0.454 0.488 0.524 0.562 0.601 0.643 0.687 0.733 0.782 0.833 0.887 0.943 1.002 1.065 1.130 1.198 1.269 1.344 1.423 1.506 1.592 1.683 1.778 1.878 1.982 2.092 2.208 2.330
168.28 169.08 169.87 170.64 171.39 172.12 172.83 173.52 174.20 174.85 175.49 176.11 176.71 177.29 177.85 178.39 178.91 179.41 179.89 180.36 180.80 181.22 181.62 182.01 182.37 182.71 183.03 183.33 183.61 183.87 184.11 184.33 184.52 184.70 184.85 184.98 185.09 185.17 185.23 185.27 185.29 185.28 185.24 185.18 185.10
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.1.
(continued) Vapor
Liquid
T(K)
P (MPa)
p (mol/dm^)
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
1.93 2.03 2.12 2.23 2.33 2.44 2.55 2.67 2.79 2.91 3.04 3.17 3.30 3.44 3.59 3.74 3.89 4.05 4.21 4.38 4.55 4.73
27.00 26.74 26.47 26.20 25.91 25.62 25.32 25.01 24.68 24.35 23.99 23.62 23.23 22.82 22.38 21.90 21.38 20.81 20.16 19.40 18.45 17.03
TABLE 7.2. T (K) 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340
267
CO (m/s)
498.21 487.88 477.36 466.63 455.69 444.51 433.10 421.42 409.47 397.21 384.62 371.63 358.17 344.14 329.41 313.80 297.06 278.88 258.79 236.08 209.33 174.74
p (mol/dm^)
CO (m/s)
2.458 2.592 2.735 2.885 3.043 3.211 3.389 3.579 3.781 3.996 4.228 4.477 4.747 5.041 5.363 5.719 6.119 6.574 7.106 7.750 8.587 9.875
184.98 184.85 184.68 184.49 184.27 184.02 183.74 183.41 183.03 182.60 182.09 181.50 180.81 180.00 179.05 177.93 176.57 174.89 172.74 169.81 165.39 157.01
Argon —Isobar at 0.101 325 MPa p (mol/dm^)
Q) (m/s)
0.140 0.125 0.113 0.103 0.095 0.088 0.082 0.077 0.072 0.068 0.064 0.061 0.058 0.056 0.053 0.051 0.049 0.047 0.045 0.044 0.042 0.041 0.039 0.038 0.037 0.036
173.81 184.14 193.74 202.78 211.38 219.59 227.49 235.09 242.44 249.56 256.48 263.21 269.76 276.15 282.39 288.50 294.48 300.33 306.07 311.70 317.24 322.67 328.02 333.27 338.45 343.55 (continues)
268
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.2. T (K) 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
(continued)
p (moydm^) 0.035 0.034 0.033 0.032 0.031 0.030 0.030 0.029 0.028 0.028 0.027 0.026 0.026 0.025 0.025 0.024 0.024 0.023 0.023 0.023 0.022 0.022 0.021 0.021 0.021 0.020 0.020 0.020 0.019 0.019 0.019 0.018 0.018 0.018 0.018 0.017
TABLE 7.3. T (K) 90 100 110 116.5981 116.5981
120 130 140 150 160
co (m/s) 348.57 353.52 358.40 363.21 367.96 372.65 377.28 381.86 386.38 390.85 395.27 399.63 403.95 408.23 412.46 416.65 420.79 424.90 428.96 432.99 436.98 440.93 444.85 448.73 452.58 456.40 460.19 463.94 467.67 471.36 475.03 478.67 482.28 485.87 489.42 492.96
Argon — Isobar at 1 MPa p (mol/dm^)
(o (m/s)
34.579 32.955 31.157 29.821 1.240 1.182 1.046 0.945 0.865 0.799
824.17 751.59 672.31 614.12 184.63 189.35 201.51 212.09 221.67 230.54
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.3. (continued) T (K)
170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
p (mol/dm^)
(o (m/s)
0.744 0.697 0.656 0.619 0.587 0.558 0.532 0.509 0.488 0.468 0.450 0.433 0.418 0.403 0.390 0.377 0.366 0.355 0.344 0.335 0.325 0.317 0.309 0.301 0.293 0.286 0.280 0.273 0.267 0.261 0.256 0.250 0.245 0.240 0.235 0.231 0.226 0.222 0.218 0.214 0.211 0.207 0.203 0.200 0.197 0.194 0.190 0.187 0.185 0.182 0.179 0.176 0.174 0.171
238.88 246.79 254.34 261.59 268.58 275.34 281.89 288.26 294.47 300.52 306.44 312.22 317.88 323.44 328.88 334.23 339.49 344.66 349.74 354.75 359.68 364.54 369.33 374.06 378.72 383.33 387.87 392.36 396.80 401.19 405.52 409.81 414.06 418.25 422.41 426.52 430.59 434.63 438.62 442.58 446.50 450.39 454.24 458.06 461.85 465.60 469.33 473.02 476.69 480.33 483.94 487.52 491.08 494.61
269
270
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.4. T (K)
90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620
Argon — Isobar at 5 MPa p (mol/dm-^)
CO (m/s)
34.879 33.343 31.682 29.828 27.655 24.831 19.159 6.090 4.947 4.307 3.867 3.534 3.269 3.051 2.865 2.706 2.566 2.443 2.332 2.232 2.142 2.060 1.984 1.914 1.849 1.789 1.734 1.681 1.632 1.586 1.543 1.502 1.464 1.427 1.392 1.359 1.328 1.298 1.270 1.242 1.216 1.191 1.167 1.145 1.123 1.101 1.081 1.061 1.043 1.024 1.007 0.990 0.973 0.958
844.78 777.34 705.93 628.46 541.00 433.54 248.19 206.39 223.58 236.49 247.45 257.20 266.11 274.40 282.20 289.60 296.66 303.44 309.98 316.29 322.42 328.37 334.17 339.82 345.34 350.74 356.03 361.22 366.31 371.31 376.23 381.06 385.82 390.51 395.13 399.69 404.18 408.62 412.99 417.32 421.59 425.81 429.99 434.12 438.20 442.25 446.25 450.21 454.13 458.02 461.87 465.68 469.46 473.21
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.4. (continued) T (K)
630 640 650 660 670 680 690 700
p (mol/dm^) 0.942 0.928 0.913 0.899 0.886 0.873 0.860 0.848
CO (m/s) 476.92 480.60 484.26 487.88 491 Al 495.04 498.58 502.09
TABLE 7.5. Argon —Isobar at 10 MPa T (K)
90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460
p (mol/dm^)
0) (m/s)
35.23 33.78 32.25 30.59 28.76 26.67 24.15 20.87 16.44 12.31 9.89 8.45 7.49 6.78 6.23 5.79 5.42 5.11 4.83 4.60 4.38 4.20 4.02 3.87 3.73 3.60 3.48 3.37 3.26 3.17 3.07 2.99 2.91 2.83 2.76 2.70 2.63 2.57
868.40 805.91 741.46 674.25 603.33 527.43 445.10 357.76 284.47 259.27 260.68 267.70 275.72 283.74 291.51 298.98 306.16 313.06 319.72 326.15 332.37 338.42 344.29 350.02 355.60 361.05 366.38 371.60 376.72 381.74 386.67 391.51 396.27 400.96 405.58 410.12 414.60 419.02 (continues)
271
272
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.5. T (K) 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
TABLE 7.6. T (K)
100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310
(continued)
p (mol/dm^) 2.51 2.46 2.41 2.36 2.31 2.26 2.22 2.18 2.13 2.10 2.06 2.02 1.99 1.95 1.92 1.89 1.86 1.83 1.80 1.77 1.75 1.72 1.70 1.67
o) (m/s) 423.38 427.69 431.94 436.14 440.29 444.39 448.45 452.46 456.43 460.36 464.25 468.10 471.91 475.69 479.44 483.15 486.83 490.47 494.09 497.67 501.23 504.76 508.26 511.73
Argon — Isobar at 50 MPa p (mol/dm^)
(0 (m/s)
36.26 35.19 34.12 33.05 31.97 30.90 29.83 28.76 27.70 26.65 25.63 24.62 23.65 22.71 21.81 20.96 20.14 19.38 18.66 17.98 17.34 16.75
969.74 927.77 887.55 849.02 812.28 777.44 744.66 714.08 685.82 659.98 636.63 615.80 597.47 581.56 567.93 556.40 546.77 538.85 532.42 527.30 523.31 520.31
TABLES O F T H E SPEED O F SOUND IN IMPORTANT FLUIDS
TABLE 7.6. T (K)
320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
TABLE 7.7. T (K) 110 120 130 140 150 160 170
(continued)
p (mol/dm^)
(0 (m/s)
16.19 15.67 15.18 14.72 14.28 13.88 13.50 13.14 12.80 12.48 12.17 11.88 11.61 11.35 11.10 10.87 10.64 10.43 10.22 10.02 9.83 9.65 9.48 9.31 9.15 9.00 8.85 8.71 8.57 8.43 8.30 8.18 8.06 7.94 7.83 7.72 7.61 7.51 7.41
518.16 516.73 515.91 515.62 515.78 516.32 517.19 518.35 519.74 521.33 523.11 525.03 527.09 529.26 531.52 533.87 536.29 538.77 541.31 543.89 546.50 549.15 551.83 554.54 557.26 560.00 562.75 565.51 568.28 571.05 573.83 576.62 579.40 582.18 584.97 587.75 590.52 593.30 596.07
. •
Argon —Isobar at 100 MPa p (mol/dm-^)
0) (m/s)
37.41 36.57 35.74 34.93 34.13 33.35 32.58
1074.29 1043.43 1014.38 987.00 961.21 936.96 914.20 (continues)
273
274
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.7. T (K) 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700
(continued)
p (mol/dm^)
co (m/s)
31.83 31.09 30.37 29.66 28.98 28.31 27.66 27.03 26.42 25.82 25.25 24.69 24.15 23.63 23.13 22.65 22.18 21.73 21.30 20.88 20.47 20.09 19.71 19.35 19.00 18.66 18.34 18.02 17.72 17.43 17.15 16.87 16.61 16.35 16.10 15.86 15.63 15.40 15.19 14.97 14.77 14.57 14.37 14.18 14.00 13.82 13.65 13.48 13.31 13.15 13.00 12.84 12.70
892.89 873.00 854.48 837.28 821.35 806.65 793.12 780.71 769.35 759.00 749.60 741.08 733.39 726.46 720.25 714.70 709.76 705.38 701.52 698.13 695.17 692.61 690.42 688.56 687.02 685.75 684.74 683.98 683.43 683.08 682.91 682.92 683.08 683.38 683.81 684.37 685.03 685.80 686.67 687.62 688.65 689.76 690.94 692.17 693.47 694.83 696.23 697.68 699.17 700.71 702.28 703.88 705.52
275
TABLES O F T H E SPEED O F SOUND IN IMPORTANT FLUIDS
TABLE 7.8.
Nitrogen — Saturation Boundary Vapor
Liquid T (K)
64 65 66 67 68 69 70 71 72 73 74 75 76 11 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115
P (MPa)
p (mol/dm^)
(i) (m/s)
p (mol/dm-^)
(D (m/s)
0.015 0.017 0.021 0.024 0.028 0.033 0.039 0.045 0.051 0.059 0.067 0.076 0.086 0.097 0.109 0.122 0.137 0.153 0.169 0.188 0.208 0.229 0.252 0.276 0.303 0.331 0.360 0.392 0.426 0.462 0.500 0.541 0.583 0.628 0.676 0.726 0.778 0.834 0.892 0.953 1.016 1.083 1.153 1.226 1.303 1.383 1.466 1.553 1.643 1.737 1.835 1.937
30.83 30.69 30.54 30.39 30.24 30.09 29.93 29.78 29.62 29.47 29.31 29.15 28.99 28.83 28.67 28.51 28.34 28.17 28.01 27.84 27.66 27.49 27.32 27.14 26.96 26.78 26.60 26.41 26.22 26.03 25.84 25.64 25.44 25.24 25.03 24.82 24.61 24.39 24.17 23.94 23.71 23.47 23.23 22.98 22.72 22.46 22.18 21.90 21.61 21.31 20.99 20.66
986.57 976.36 966.18 956.04 945.93 935.83 925.74 915.66 905.58 895.49 885.39 875.28 865.15 855.00 844.82 834.61 824.36 814.07 803.74 793.36 782.93 772.44 761.89 751.28 740.60 729.84 719.01 708.09 697.09 685.99 674.80 663.50 652.09 640.57 628.92 617.14 605.23 593.17 580.96 568.58 556.03 543.30 530.37 517.24 503.88 490.29 476.44 462.32 447.92 433.19 418.12 402.67
0.028 0.033 0.038 0.044 0.051 0.059 0.068 0.077 0.088 0.100 0.112 0.126 0.142 0.158 0.176 0.196 0.217 0.240 0.265 0.292 0.320 0.351 0.383 0.418 0.456 0.496 0.538 0.584 0.632 0.683 0.737 0.795 0.856 0.921 0.990 1.063 1.141 1.223 1.310 1.403 1.501 1.605 1.716 1.833 1.958 2.092 2.234 2.386 2.549 2.724 2.912 3.116
162.08 163.20 164.30 165.37 166.41 167.43 168.42 169.39 170.32 171.23 172.10 172.95 173.77 174.55 175.31 176.03 176.72 177.38 178.00 178.60 179.16 179.68 180.17 180.63 181.05 181.43 181.78 182.10 182.38 182.62 182.82 182.99 183.12 183.21 183.26 183.28 183.25 183.18 183.08 182.93 182.74 182.51 182.24 181.93 181.58 181.19 180.76 180.28 179.75 179.15 178.49 177.75 (continues)
276
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.8.
{continued)
Liquid T (K)
116 117 118 119 120 121 122 123 124 125 126
P (MPa)
p (mol/dm-^)
2.043 2.153 2.268 2.387 2.511 2.639 2.773 2.912 3.056 3.207 3.365
20.31 19.94 19.55 19.13 18.68 18.19 17.63 17.00 16.23 15.21 13.28
TABLE 7.9.
Vapor CO (m/s) 386.80 370.43 353.49 335.85 317.33 297.68 276.54 253.32 227.00 195.48 150.97
p (mol/dm-^)
0) (m/s)
3.337 3.579 3.844 4.137 4.465 4.838 5.270 5.785 6.430 7.324 9.111
176.93 176.01 175.00 173.87
Nitrogen —Isobar at 0.101 325 MPa
T (K)
p (mol/dm^)
(D (m/s)
70
29.94 28.77 0.1646 0.1392 0.1125 0.0946 0.0817 0.0720 0.0643 0.0581 0.0530 0.0488 0.0452 0.0420 0.0393 0.0369 0.0348 0.0329 0.0312 0.0297 0.0283 0.0271 0.0259 0.0249 0.0239 0.0230 0.0221 0.0214 0.0206 0.0200 0.0193 0.0187 0.0182
926.19 851.39 174.82 190.42 212.07 231.36 248.99 265.37 280.74 295.28 309.11 322.33 335.02 347.22 359.00 370.38 381.40 392.08 402.44 412.51 422.31 431.84 441.12 450.16 458.99 467.60 476.02 484.25 492.31 500.20 507.93 515.51 522.95
77.355 77.355
90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670
\11.6\ 171.17 169.49 167.43 164.67 160.26 148.38
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.9. T (K) 690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990 1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730
(continued)
p (moydm^) 0.0177 0.0172 0.0167 0.0162 0.0158 0.0154 0.0150 0.0147 0.0143 0.0140 0.0137 0.0134 0.0131 0.0128 0.0126 0.0123 0.0121 0.0118 0.0116 0.0114 0.0112 0.0110 0.0108 0.0106 0.0104 0.0102 0.0101 0.0099 0.0097 0.0096 0.0094 0.0093 0.0092 0.0090 0.0089 0.0088 0.0086 0.0085 0.0084 0.0083 0.0082 0.0081 0.0080 0.0079 0.0078 0.0077 0.0076 0.0075 0.0074 0.0073 0.0072 0.0071 0.0070
co (m/s) 530.27 537.45 544.53 551.49 558.34 565.10 571.76 578.33 584.82 591.22 597.55 603.80 609.99 616.10 622.15 628.14 634.07 639.94 645.75 651.51 657.22 662.87 668.48 674.04 679.56 685.03 690.45 695.84 701.18 706.48 711.75 716.97 722.16 727.31 732.43 737.51 742.56 747.57 752.56 757.51 762.43 767.31 772.17 777.00 781.80 786.57 791.31 796.03 800.72 805.38 810.02 814.63 819.21 (continues)
277
278
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
TABLE 7.9. (continued) T (K)
p (mol/dm^) 0.0070 0.0069 0.0068 0.0067 0.0067 0.0066 0.0065 0.0064 0.0064 0.0063 0.0062 0.0062 0.0061
1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
CO (m/s) 823.78 828.31 832.82 837.31 841.78 846.22 850.64 855.04 859.42 863.77 868.10 872.42 876.71
TABLE 7.10. Nitrogen — Isobar at 1 MPa T (K)
p (mol/dm^)
0) (m/s)
70 90
30.00 26.69 23.77 1.48 1.32 1.02 0.8523 0.7359 0.6498 0.5829 0.5291 0.4848 0.4475 0.4158 0.3883 0.3643 0.3432 0.3244 0.3076 0.2925 0.2788 0.2663 0.2549 0.2445 0.2349 0.2260 0.2178 0.2101 0.2030 0.1963 0.1901 0.1843
932.51 727.06 559.22 182.79 194.09 221.53 243.20 261.98 278.91 294.53 309.13 322.92 336.03 348.56 360.58 372.16 383.33 394.13 404.58 414.73 424.57 434.15 443.46 452.53 461.37 469.99 478.42 486.65 494.70 502.58 510.30 517.88
103.747 103.747
110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.10.
(continued)
T (K)
p (mol/dm^)
(0 (m/s)
670 690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990
0.1788 0.1736 0.1687 0.1641 0.1597 0.1556 0.1516 0.1479 0.1443 0.1410 0.1377 0.1346 0.1317 0.1289 0.1262 0.1236 0.1211 0.1187 0.1164 0.1142 0.1120 0.1100 0.1080 0.1061 0.1043 0.1025 0.1008 0.0991 0.0975 0.0959 0.0944 0.0930 0.0916 0.0902 0.0888 0.0876 0.0863 0.0851 0.0839 0.0827 0.0816 0.0805 0.0795 0.0784 0.0774 0.0764 0.0755 0.0745 0.0736 0.0727 0.0719 0.0710 0.0702
525.31 532.61 539.79 546.84 553.79 560.63 567.37 574.01 580.57 587.04 593.43 599.74 605.98 612.14 618.24 624.27 630.25 636.16 642.01 647.81 653.55 659.24 664.89 670.48 676.02 681.52 686.98 692.39 697.76 703.09 708.38 713.63 718.84 724.02 729.16 734.26 739.33 744.37 749.37 754.34 759.28 764.18 769.06 773.91 778.72 783.51 788.27 793.00 797.71 802.39 807.04 811.67 816.27
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710
(continues)
279
280
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.10. T (K) 1730 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990 2010
TABLE 7.11.
(continued)
p (mol/dnr) 0.0694 0.0686 0.0678 0.0670 0.0663 0.0656 0.0649 0.0642 0.0635 0.0628 0.0622 0.0616 0.0609 0.0603 0.0597
CO (m/s)
820.84 825.39 829.92 834.42 838.90 843.36 847.79 852.20 856.59 860.96 865.31 869.63 873.94 878.22 882.48
Nitrogen — Isobar at 5 MPa
T (K)
p (mol/dm^)
CO (m/s)
70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650
30.283 27.202 23.385 16.433 6.029 4.387 3.605 3.108 2.753 2.483 2.266 2.089 1.940 1.812 1.702 1.605 1.519 1.442 1.373 1.311 1.254 1.202 1.154 1.111 1.070 1.032 0.997 0.965 0.934 0.906
959.16 772.11 565.77 288.81 226.09 254.67 277.24 296.46 313.52 329.07 343.47 356.96 369.71 381.84 393.42 404.54 415.24 425.55 435.52 445.18 454.54 463.63 472.48 481.09 489.48 497.67 505.67 513.49 521.15 528.65
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.11.
(continued)
T (K)
p (mol/dwr)
0) (m/s)
670 690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990
0.879 0.853 0.830 0.807 0.786 0.765 0.746 0.728 0.711 0.694 0.678 0.663 0.649 0.635 0.622 0.609 0.597 0.585 0.574 0.563 0.553 0.543 0.533 0.524 0.515 0.506 0.498 0.490 0.482 0.474 0.467 0.460 0.453 0.446 0.439 0.433 0.427 0.421 0.415 0.409 0.404 0.399 0.393 0.388 0.383 0.378 0.374 0.369 0.365 0.360 0.356 0.352 0.348
536.01 543.24 550.33 557.31 564.17 570.93 577.59 584.15 590.62 597.01 603.32 609.55 615.70 621.79 627.81 633.77 639.66 645.49 651.27 657.00 662.67 668.29 673.86 679.38 684.86 690.29 695.68 701.02 706.33 711.60 716.82 722.01 727.16 732.28 737.36 742.40 747.42 752.40 757.34 762.26 767.14 772.00 776.82 781.62 786.38 791.12 795.83 800.52 805.17 809.81 814.41 818.99 823.55
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710
(continues)
281
282
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.11.
(continued)
T (K)
p (mol/dm^)
1730 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
0.344 0.340 0.336 0.332 0.329 0.325 0.322 0.318 0.315 0.312 0.309 0.305 0.302 0.299
TABLE 7.12.
CO (m/s) 828.08 832.59 837.07 841.53 845.97 850.38 854.78 859.15 863.50 867.82 872.13 876.42 880.68 884.93
Nitrogen —Isobar at 10 MPa
T (K)
p (mol/dm^)
0) (m/s)
70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670
30.60 27.75 24.49 20.41 14.96 10.28 7.87 6.53 5.65 5.02 4.54 4.15 3.84 3.57 3.35 3.15 2.98 2.82 2.69 2.56 2.45 2.35 2.26 2.17 2.09 2.02 1.95 1.89 1.83 1.77 1.72
989.62 819.44 647.61 472.34 331.25 292.70 300.47 315.42 330.91 345.77 359.83 373.13 385.75 397.78 409.27 420.29 430.88 441.09 450.94 460.47 469.71 478.67 487.38 495.85 504.10 512.15 520.01 527.69 535.21 542.57 549.80
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.12. T (K)
690 710 730 750 770 790 810 830 850 870 890 910 930 950 970 990 1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730
(continued)
p (mol/dm-^) 1.67 1.62 1.58 1.54 1.50 1.46 1.43 1.39 1.36 1.33 1.30 1.27 1.25 1.22 1.20 1.17 1.15 1.13 1.11 1.09 1.07 1.05 1.03 1.01 1.00 0.981 0.965 0.950 0.935 0.920 0.906 0.893 0.880 0.867 0.855 0.843 0.831 0.820 0.809 0.798 0.787 0.777 0.767 0.758 0.748 0.739 0.730 0.721 0.713 0.704 0.696 0.688 0.680
(D (m/s) 556.88 563.84 570.69 577.42 584.05 590.58 597.01 603.36 609.63 615.82 621.93 627.97 633.95 639.86 645.71 651.49 657.23 662.90 668.53 674.10 679.62 685.10 690.53 695.92 701.26 706.56 711.82 717.05 722.23 727.37 732.48 737.56 742.59 747.60 752.57 757.51 762.42 767.29 772.14 776.96 781.74 786.50 791.23 795.93 800.61 805.26 809.88 814.48 819.05 823.60 828.12 832.62 837.10 {continues)
283
284
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.12. T (K) 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
TABLE 7.13.
(continued)
p (mol/dm^) 0.673 0.665 0.658 0.651 0.644 0.637 0.630 0.624 0.617 0.611 0.605 0.599 0.593
CO (m/s) 841.55 845.98 850.38 854.77 859.13 863.47 867.79 872.09 876.37 880.63 884.87 889.09 893.29
Nitrogen —Isobar at 50 MPa
T (K)
p (mol/dm^)
(0 (m/s)
90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670 690 710
30.52 28.54 26.61 24.76 22.99 21.35 19.83 18.45 17.22 16.12 15.14 14.28 13.50 12.81 12.19 11.64 11.13 10.67 10.25 9.87 9.51 9.18 8.88 8.60 8.33 8.08 7.85 7.64 7.43 7.24 7.05 6.88
1061.02 963.85 878.83 806.60 747.47 701.00 665.92 640.44 622.64 610.73 603.25 599.08 597.38 597.50 598.98 601.45 604.67 608.44 612.61 617.08 621.78 626.63 631.59 636.63 641.73 646.85 651.99 657.14 662.28 667.41 672.53 677.63
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.13. (continued) T (K)
p (mol/dm^)
(0 (m/s)
730 750 770 790 810 830 850 870 890 910 930 950 970 990
6.72 6.56 6.41 6.27 6.14 6.01 5.88 5.77 5.65 5.54 5.44 5.34 5.24 5.15 5.06 4.97 4.89 4.81 4.73 4.66 4.59 4.51 4.45 4.38 4.32 4.25 4.19 4.13 4.08 4.02 3.97 3.92 3.87 3.82 3.77 3.72 3.67 3.63 3.59 3.54 3.50 3.46 3.42 3.38 3.34 3.31 3.27 3.24 3.20 3.17 3.13 3.10 3.07
682.70 687.76 692.79 697.80 702.78 707.74 712.67 717.58 722.46 727.31 732.14 736.95 741.73 746.49 751.22 755.93 760.62 765.28 769.92 774.54 779.14 783.71 788.26 792.79 797.30 801.79 806.26 810.70 815.13 819.54 823.92 828.29 832.64 836.96 841.27 845.56 849.83 854.09 858.32 862.54 866.73 870.91 875.08 879.22 883.35 887.46 891.56 895.63 899.70 903.74 907.77 911.78 915.78
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730 1750 1770
(continues)
285
286
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.13.
(continued)
T (K)
p (mol/dm-^)
CO (m/s)
1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990
3.04 3.01 2.98 2.95 2.92 2.89 2.86 2.84 2.81 2.78 2.76
919.76 923.73 927.68 931.61 935.53 939.44 943.33 947.20 951.06 954.91 958.74
TABLE 7.14. T (K)
90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 510 530 550 570 590 610 630 650 670 690 710 730 750
Nitrogen —at 100 MPa p (mol/dm^)
CO (m/s)
32.57 31.03 29.57 28.19 26.90 25.69 24.57 23.52 22.54 21.63 20.78 19.99 19.26 18.58 17.94 17.35 16.79 16.27 15.79 15.33 14.90 14.49 14.11 13.75 13.41 13.09 12.78 12.49 12.21 11.95 11.69 11.45 11.22 11.00
1244.55 1173.11 1110.14 1055.27 1008.09 967.97 934.20 906.03 882.74 863.64 848.09 835.56 825.55 817.66 811.52 806.84 803.39 800.95 799.36 798.49 798.21 798.44 799.10 800.12 801.46 803.07 804.91 806.95 809.16 811.53 814.04 816.67 819.40 822.23
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.14. (continued) T (K)
p (mol/dm^)
CO (m/s)
770 790 810 830 850 870 890 910 930 950 970 990
10.79 10.58 10.39 10.20 10.02 9.85 9.68 9.52 9.36 9.21 9.07 8.93 8.79 8.66 8.53 8.41 8.29 8.17 8.06 7.95 7.84 7.74 7.64 7.54 7.44 7.35 7.26 7.17 7.08 7.00 6.92 6.84 6.76 6.68 6.61 6.54 6.46 6.39 6.33 6.26 6.19 6.13 6.07 6.01 5.95 5.89 5.83 5.77 5.72 5.66 5.61 5.56 5.51
825.14 828.12 831.18 834.29 837.46 840.68 843.94 847.25 850.58 853.95 857.35 860.78 864.23 867.69 871.18 874.68 878.20 881.73 885.27 888.82 892.38 895.95 899.52 903.10 906.68 910.26 913.84 917.43 921.01 924.60 928.18 931.77 935.35 938.93 942.50 946.07 949.64 953.21 956.76 960.32 963.87 967.41 970.95 974.48 978.01 981.53 985.04 988.55 992.05 995.55 999.03 1002.51 1005.98
1010 1030 1050 1070 1090 1110 1130 1150 1170 1190 1210 1230 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650 1670 1690 1710 1730 1750 1770 1790 1810
(continues)
287
288
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.14.
(continued)
T (K)
p (mol/dm^)
CO (m/s] 1
1830 1850 1870 1890 1910 1930 1950 1970 1990
5.46 5.41 5.36 5.31 5.27 5.22 5.18 5.13 5.09
1009.45 1012.91 1016.36 1019.8C1 1023.23 1026.66 1030.08 1033.491 1036.9C»
T A B L E 7.15.
Water—Saturation Boundary Vapor
Liquid T (K)
275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343
P (MPa)
p (mol/dm^)
0) (m/s)
p (mol/dm^)
0.0007 0.0008 0.0009 0.0011 0.0012 0.0014 0.0016 0.0018 0.0020 0.0023 0.0026 0.0030 0.0033 0.0038 0.0042 0.0047 0.0053 0.0059 0.0066 0.0073 0.0081 0.0090 0.0100 0.0111 0.0123 0.0135 0.0149 0.0164 0.0180 0.0198 0.0217 0.0238 0.0260 0.0284 0.031
55.50 55.50 55.50 55.50 55.49 55.48 55.47 55.45 55.43 55.41 55.38 55.36 55.33 55.30 55.27 55.23 55.20 55.16 55.12 55.08 55.03 54.99 54.94 54.90 54.85 54.80 54.74 54.69 54.63 54.58 54.52 54.46 54.40 54.34 54.28
1411.35 1420.78 1429.78 1438.35 1446.51 1454.29 1461.68 1468.71 1475.39 1481.72 1487.72 1493.41 1498.78 1503.86 1508.64 1513.14 1517.37 1521.33 1525.04 1528.49 1531.70 1534.68 1537.42 1539.95 1542.25 1544.35 1546.24 1547.93 1549.43 1550.73 1551.85 1552.80 1553.56 1554.16 1554.59
0.0003 0.0003 0.0004 0.0005 0.0005 0.0006 0.0007 0.0008 0.0008 0.0010 0.0011 0.0012 0.0013 0.0015 0.0017 0.0019 0.0021 0.0023 0.0026 0.0028 0.0031 0.0034 0.0038 0.0042 0.0046 0.0050 0.0055 0.0060 0.0066 0.0072 0.0078 0.0085 0.0093 0.010 0.011
CO (m/s)
410.33 411.77 413.21 414.64 416.06 417.48 418.89 420.29 421.69 423.08 424.46 425.84 427.21 428.57 429.93 431.28 432.62 433.96 435.29 436.61 437.93 439.24 440.54 441.83 443.11 444.39 445.66 446.92 448.17 449.41 450.64 451.87 453.08 454.28 455.48
289
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.15. (continued) Vapor
Liquid T (K)
345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445 447
P (MPa) 0.034 0.037 0.040 0.043 0.047 0.051 0.055 0.060 0.065 0.070 0.075 0.081 0.087 0.094 0.101 0.108 0.116 0.125 0.133 0.143 0.153 0.163 0.174 0.186 0.198 0.211 0.224 0.238 0.253 0.269 0.286 0.303 0.321 0.340 0.360 0.381 0.403 0.426 0.449 0.474 0.500 0.527 0.556 0.585 0.616 0.648 0.681 0.716 0.752 0.789 0.828 0.869
p
{mol/dw?) 54.21 54.15 54.08 54.01 53.95 53.88 53.81 53.73 53.66 53.59 53.51 53.44 53.36 53.28 53.20 53.12 53.04 52.96 52.88 52.79 52.71 52.62 52.53 52.45 52.36 52.27 52.18 52.08 51.99 51.90 51.80 51.71 51.61 51.51 51.42 51.32 51.22 51.11 51.01 50.91 50.81 50.70 50.59 50.49 50.38 50.27 50.16 50.05 49.94 49.82 49.71 49.60
0) (m/s)
1554.85 1554.96 1554.91 1554.71 1554.36 1553.86 1553.22 1552.44 1551.52 1550.47 1549.29 1547.98 1546.54 1544.97 1543.29 1541.48 1539.56 1537.52 1535.36 1533.09 1530.71 1528.22 1525.62 1522.92 1520.11 1517.20 1514.18 1511.06 1507.85 1504.53 1501.12 1497.61 1494.00 1490.30 1486.51 1482.62 1478.64 1474.57 1470.41 1466.17 1461.83 1457.40 1452.89 1448.29 1443.60 1438.83 1433.97 1429.03 1424.01 1418.90 1413.71 1408.44
p
{moVdw?) 0.012 0.013 0.014 0.015 0.016 0.017 0.019 0.020 0.022 0.023 0.025 0.027 0.029 0.031 0.033 0.035 0.038 0.040 0.043 0.046 0.049 0.052 0.055 0.058 0.062 0.066 0.070 0.074 0.078 0.083 0.087 0.092 0.098 0.103 0.109 0.115 0.121 0.127 0.134 0.141 0.148 0.156 0.164 0.172 0.180 0.189 0.198 0.208 0.218 0.228 0.239 0.250
0) (m/s)
456.67 457.84 459.01 460.16 461.30 462.44 463.56 464.67 465.77 466.86 467.93 469.00 470.05 471.09 472.12 473.13 474.14 475.12 476.10 477.06 478.01 478.95 479.87 480.77 481.67 482.54 483.41 484.25 485.09 485.91 486.71 487.50 488.27 489.02 489.76 490.49 491.19 491.88 492.56 493.22 493.86 494.48 495.09 495.68 496.25 496.81 497.35 497.87 498.37 498.86 499.32 499.77 (continues)
290
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.15.
(continued)
Liquid T (K)
449 451 453 455 457 459 461 463 465 467 469 471 473 475 477 479 481 483 485 487 489 491 493 495 497 499 501 503 505 507 509 511 513 515 517 519 521 523 525 527 529 531 533 535 537 539 541 543 545 547
P (MPa) 0.911 0.954 0.999 1.05 1.09 1.15 1.20 1.25 1.31 1.36 1.42 1.49 1.55 1.62 1.68 1.75 1.83 1.90 1.98 2.06 2.14 2.23 2.31 2.40 2.50 2.59 2.69 2.79 2.89 3.00 3.11 3.22 3.34 3.46 3.58 3.70 3.83 3.97 4.10 4.24 4.38 4.53 4.68 4.83 4.99 5.15 5.32 5.49 5.66 5.84
Vapor
p (mol/dm-^)
0) (m/s)
p (mol/dm-^)
CO (m/s)
49.48 49.36 49.24 49.13 49.01 48.88 48.76 48.64 48.51 48.39 48.26 48.13 48.01 47.88 47.74 47.61 47.48 47.34 47.21 47.07 46.93 46.79 46.65 46.51 46.36 46.22 46.07 45.92 45.77 45.62 45.47 45.32 45.16 45.00 44.84 44.68 44.52 44.36 44.19 44.02 43.86 43.68 43.51 43.34 43.16 42.98 42.80 42.61 42.43 42.24
1403.08 1397.64 1392.12 1386.52 1380.84 1375.08 1369.24 1363.32 1357.32 1351.24 1345.08 1338.84 1332.53 1326.13 1319.66 1313.10 1306.47 1299.76 1292.98 1286.11 1279.16 1272.14 1265.04 1257.85 1250.59 1243.25 1235.83 1228.33 1220.76 1213.10 1205.36 1197.53 1189.63 1181.65 1173.58 1165.43 1157.19 1148.88 1140.47 1131.98 1123.41 1114.74 1105.99 1097.15 1088.21 1079.19 1070.07 1060.86 1051.55 1042.15
0.261 0.273 0.285 0.298 0.311 0.325 0.339 0.354 0.369 0.385 0.401 0.418 0.435 0.453 0.471 0.491 0.510 0.531 0.552 0.573 0.596 0.619 0.643 0.668 0.693 0.719 0.746 0.774 0.803 0.833 0.863 0.895 0.927 0.961 0.995 1.031 1.068 1.105 1.144 1.185 1.226 1.269 1.313 1.358 1.405 1.453 1.503 1.554 1.607 1.662
500.20 500.62 501.01 501.39 501.74 502.08 502.40 502.70 502.98 503.24 503.49 503.71 503.91 504.09 504.25 504.39 504.51 504.61 504.68 504.74 504.77 504.78 504.77 504.74 504.68 504.60 504.49 504.36 504.21 504.03 503.82 503.59 503.34 503.06 502.75 502.41 502.05 501.65 501.23 500.78 500.30 499.79 499.25 498.68 498.07 497.43 496.76 496.06 495.32 494.54
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.15.
(continued) Vapor
Liquid T (K)
P (MPa)
549 551 553 555 557 559 561 563 565 567 569 571 573 575 577 579 581 583 585 587 589 591 593 595 597 599 601 603 605 607 609 611 613 615 617 619 621 623 625 627 629 631 633 635 637 639 641 643 645 647
6.02 6.21 6.40 6.60 6.80 7.00 7.21 7.43 7.64 7.87 8.10 8.33 8.57 8.81 9.06 9.32 9.58 9.84 10.12 10.39 10.68 10.97 11.26 11.56 11.87 12.19 12.51 12.83 13.17 13.51 13.86 14.21 14.57 14.94 15.32 15.71 16.10 16.50 16.91 17.33 17.75 18.19 18.63 19.09 19.55 20.02 20.51 21.01 21.52 22.04
291
p (mol/dm^)
CO (m/s)
42.05 41.86 41.66 41.46 41.26 41.06 40.85 40.64 40.43 40.21 40.00 39.77 39.55 39.32 39.08 38.85 38.60 38.36 38.11 37.85 37.59 37.32 37.05 36.77 36.49 36.20 35.90 35.59 35.28 34.95 34.62 34.28 33.92 33.56 33.18 32.78 32.37 31.93 31.48 31.00 30.48 29.93 29.33 28.68 27.97 27.17 26.26 25.15 23.59 19.84
1032.65 1023.04 1013.34 1003.53 993.61 983.59 973.46 963.21 952.85 942.38 931.78 921.06 910.22 899.25 888.14 876.90 865.53 854.00 842.34 830.52 818.54 806.40 794.09 781.60 768.93 756.07 743.01 729.72 716.21 702.44 688.40 674.06 659.38 644.33 628.86 612.91 596.41 579.28 561.45 542.82 523.31 502.83 481.38 459.03 435.99 412.53 388.55 362.10 325.47 251.20
p (mol/dm^) 1.718 1.776 1.836 1.898 1.962 2.028 2.096 2.167 2.240 2.315 2.393 2.473 2.556 2.643 2.732 2.824 2.920 3.020 3.123 3.230 3.342 3.458 3.579 3.705 3.836 3.973 4.116 4.265 4.422 4.587 4.760 4.942 5.134 5.337 5.552 5.781 6.025 6.286 6.566 6.869 7.198 7.558 7.956 8.401 8.907 9.496 10.21 11.12 12.46 15.90
CO (m/s)
493.73 492.88 491.99 491.07 490.10 489.10 488.05 486.96 485.83 484.65 483.42 482.15 480.83 479.46 478.04 476.56 475.03 473.44 471.80 470.09 468.32 466.48 464.58 462.60 460.55 458.42 456.21 453.91 451.52 449.04 446.45 443.75 440.93 437.99 434.90 431.67 428.26 424.67 420.87 416.82 412.49 407.83 402.77 397.19 390.93 383.74 375.08 363.84 346.60 285.31
292
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.16. T (K) 275 295 315 335 355 373.12 373.12
375 395 415 435 455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
Water—Isobar at 0.101 325 MPa p (mol/dm-^) 55.51 55.39 55.04 54.52 53.88 53.20 0.0332 0.0330 0.0312 0.0296 0.0282 0.0269 0.0258 0.0247 0.0238 0.0229 0.0220 0.0212 0.0205 0.0199 0.0192 0.0186 0.0181 0.0176 0.0171 0.0166 0.0162 0.0157 0.0153 0.0150 0.0146 0.0143 0.0139 0.0136 0.0133 0.0130 0.0128 0.0125 0.0123 0.0120 0.0118 0.0116 0.0113 0.0111 0.0109 0.0107 0.0106 0.0104 0.0102 0.0100 0.0099 0.0097
co (m/s) 1411.51 1487,89 1531.87 1552.00 1553.96 1543.18 472.18 473.52 487.03 499.70 511.80 523.44 534.70 545.62 556.23 566.58 576.68 586.55 596.21 605.68 614.95 624.05 632.98 641.75 650.37 658.84 667.18 675.39 683.47 691.43 699.27 707.00 714.63 722.16 729.59 736.92 744.17 751.33 758.41 765.41 772.33 779.17 785.95 792.66 799.30 805.88 812.40 818.86 825.26 831.61 837.90 844.14
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.17.
Water—isobar at 1 MPa
T (K)
p (mol/dm^)
275 295 315 335 355 375 395 415 435 453.03 453.03
455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
55.53 55.41 55.06 54.54 53.90 53.15 52.29 51.34 50.28 49.24 0.286 0.284 0.268 0.255 0.243 0.232 0.223 0.214 0.207 0.199 0.193 0.186 0.181 0.175 0.170 0.165 0.161 0.156 0.152 0.148 0.145 0.141 0.138 0.135 0.132 0.129 0.126 0.124 0.121 0.119 0.116 0.114 0.112 0.110 0.108 0.106 0.104 0.102 0.101 0.099 0.097 0.096
0) (m/s)
1412.95 1489.38 1533.44 1553.68 1555.76 1543.41 1519.06 1484.24 1439.85 1392.05 501.02 502.85 518.73 532.48 545.19 557.16 568.56 579.48 590.01 600.2 610.09 619.72 629.11 638.29 647.26 656.05 664.67 673.12 681.42 689.58 697.61 705.51 713.29 720.95 728.50 735.95 743.30 750.56 757.72 764.80 771.80 778.72 785.56 792.33 799.03 805.66 812.22 818.73 825.17 831.56 837.89 844.16
293
294
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.18.
Water—Isobar at 5 MPa
T (K)
p (mol/dm^)
CO (m/s)
55.64 55.51 55.15 54.64 54.00 53.25 52.40 51.46 50.41 49.27 48.02 46.64 45.09 43.35 43.15 1.41 1.30 1.22 1.15 1.09 1.04 1.00 0.956 0.921 0.889 0.859 0.832 0.807 0.784 0.762 0.741 0.722 0.704 0.686 0.670 0.655 0.640 0.626 0.612 0.600 0.587 0.576 0.564 0.554 0.543 0.533 0.524 0.515 0.506 0.497 0.489 0.481
1419.38 1496.03 1540.44 1561.11 1563.72 1552.01 1528.44 1494.56 1451.34 1399.29 1338.56 1268.87 1189.36 1098.15 1087.81 498.04 520.81 540.41 557.18 572.16 585.84 598.56 610.51 621.84 632.65 643.03 653.03 662.71 672.09 681.21 690.09 698.76 707.23 715.52 723.65 731.62 739.45 747.14 754.7 762.15 769.48 776.72 783.85 790.88 797.83 804.69 811.47 818.18 824.81 831.37 837.86 844.29
275 295 315 335 355 375 395 415 435 455 475 495 515 535 537.09 537.09
555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.19.
Water—Isobar at 10 MPa
T (K)
p (mol/dm^)
(D (m/s)
55.78 55.63 55.27 54.76 54.12 53.38 52.54 51.60 50.58 49.45 48.23 46.88 45.38 43.70 41.78 39.47 38.21 3.08 2.85 2.56 2.36 2.21 2.09 1.99 1.90 1.82 1.75 1.69 1.63 1.58 1.53 1.48 1.44 1.40 1.37 1.33 1.30 1.27 1.24 1.21 1.19 1.16 1.14 1.12 1.09 1.07 1.05 1.03 1.02 1.00 0.98 9.95
1427.48 1504.34 1549.15 1570.33 1573.56 1562.62 1539.98 1507.23 1465.38 1415.01 1356.38 1289.39
275 295 315 335 355 375 395 415 435 455 475 495 515 535 555 575 584.15 584.15
595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995 1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
nn.Ai 1127.35 1028.37 910.76 847.33 472.51 494.96 524.55 547.42 566.64 583.48 598.63 612.52 625.42 637.54 649.02 659.95 670.42 680.49 690.21 699.62 708.74 717.62 726.27 734.71 742.96 751.03 758.95 766.71 774.33 781.83 789.2 796.46 803.60 810.65 817.60 824.46 831.24 837.93 844.54
295
296
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.20.
Water—Isobar at 50 MPa
T (K)
p (mol/dm^)
0) (m/s)
275 295 315 335 355 375 395 415 435 455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995
56.82 56.59 56.19 55.67 55.05 54.34 53.56 52.70 51.77 50.76 49.69 48.53 47.29 45.96 44.52 42.95 41.23 39.32 37.17 34.70 31.78 28.24 24.09 19.85 16.46 14.09 12.44 11.24 10.34 9.62 9.04 8.55 8.13 7.77 7.45 7.17 6.91 6.68 6.47 6.28 6.10 5.94 5.78 5.64 5.51 5.38 5.26 5.15 5.04 4.94
1494.62 1570.91 1617.39 1641.44 1648.59 1642.61 1626.02 1600.54 1567.36 1527.34 1481.12 1429.22 1372.03 1309.84 1242.87 1171.23 1094.94 1013.97 928.41 838.86 746.93 659.20 590.51 556.44 556.75 571.69 590.63 609.70 627.78 644.65 660.37 675.07 688.87 701.91 714.26 726.03 737.29 748.08 758.46 768.47 778.15 787.53 796.64 805.49 814.11 822.53 830.74 838.77 846.64 854.34
1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.21.
Water—Isobar at 100 MPa
T (K)
p (mol/dm^)
(0 (m/s)
275 295 315 335 355 375 395 415 435 455 475 495 515 535 555 575 595 615 635 655 675 695 715 735 755 775 795 815 835 855 875 895 915 935 955 975 995
58.00 57.67 57.24 56.71 56.10 55.43 54.69 53.90 53.05 52.14 51.18 50.17 49.11 47.99 46.81 45.58 44.27 42.90 41.45 39.92 38.31 36.62 34.84 32.99 31.08 29.15 27.24 25.40 23.67 22.08 20.65 19.37 18.24 17.24 16.37 15.59 14.90 14.28 13.72 13.22 12.76 12.35 11.96 11.61 11.28 10.98 10.70 10.44 10.19 9.95
1583.41 1654.36 1699.71 XllASA 1734.86 1732.90 1721.44 1702.11 1676.13 1644.40 1607.67 1566.57 1521.67 1473.50 1422.55 1369.34 1314.37 1258.17 1201.31 1144.39 1088.12 1033.35 981.04 932.30 888.40 850.76 820.40 797.37 781.16 771.04 766.04 765.04 767.02 771.13 776.73 783.33 790.61 798.30 806.26 814.35 822.49 830.64 838.74 846.78 854.73 862.60 870.36 878.01 885.56 893.00
1015 1035 1055 1075 1095 1115 1135 1155 1175 1195 1215 1235 1255
297
298
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.22.
Air—Saturation Boundary
T (K)
P (Bubble) (MPa)
Liquid p (mol/dm^)
CO (m/s)
P (Dew) (MPa)
Vapor p (mol/dm^)
CO (m/s)
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
0.006 0.007 0.008 0.010 0.012 0.014 0.017 0.020 0.024 0.027 0.032 0.037 0.042 0.049 0.056 0.063 0.072 0.081 0.091 0.10 0.11 0.13 0.14 0.16 0.17 0.19 0.21 0.23 0.26 0.28 0.30 0.33 0.36 0.39 0.42 0.46 0.50 0.53 0.57 0.62 0.66 0.71 0.76 0.81 0.87 0.93 0.99 1.05 1.12 1.18 1.26 1.33 1.41 1.49 1.58 1.66 1.75
33.03 32.89 32.74 32.60 32.46 32.31 32.17 32.02 31.87 31.72 31.58 31.43 31.28 31.13 30.97 30.82 30.67 30.51 30.36 30.20 30.04 29.88 29.72 29.56 29.40 29.23 29.07 28.90 28.73 28.56 28.38 28.21 28.03 27.85 27.67 27.49 27.30 27.12 26.92 26.73 26.53 26.33 26.13 25.92 25.71 25.50 25.28 25.06 24.83 24.60 24.36 24.12 23.87 23.61 23.35 23.08 22.80
1028.29 1020.26 1012.16 1004.00 995.77 987.48 979.13 970.72 962.24 953.70 945.10 936.43 927.70 918.90 910.04 901.11 892.11 883.05 873.91 864.71 855.44 846.09 836.67 827.18 817.61 807.96 798.24 788.44 778.56 768.59 758.55 748.42 738.20 727.90 717.51 707.03 696.46 685.80 675.05 664.20 653.26 642.22 631.08 619.84 608.50 597.06 585.51 573.85 562.09 550.21 538.21 526.10 513.86 501.48 488.97 476.31 463.48
0.0026 0.0033 0.0041 0.0051 0.0063 0.0078 0.0094 0.011 0.014 0.016 0.019 0.023 0.027 0.031 0.037 0.042 0.049 0.056 0.064 0.073 0.082 0.093 0.10 0.12 0.13 0.15 0.16 0.18 0.20 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.45 0.49 0.53 0.57 0.61 0.66 0.71 0.76 0.81 0.87 0.93 0.99 1.06 1.13 1.20 1.28 1.35 1.44 1.52 1.61
0.005 0.006 0.008 0.010 0.012 0.014 0.017 0.021 0.024 0.029 0.034 0.039 0.046 0.053 0.061 0.069 0.079 0.090 0.101 0.114 0.128 0.143 0.160 0.178 0.198 0.219 0.241 0.266 0.292 0.320 0.351 0.383 0.417 0.454 0.493 0.535 0.580 0.627 0.677 0.730 0.786 0.846 0.909 0.976 1.047 1.122 1.201 1.285 1.374 1.468 1.568 1.674 1.786 1.905 2.032 2.166 2.310
155.14 156.38 157.61 158.81 159.99 161.16 162.30 163.42 164.53 165.61 166.66 167.70 168.70 169.69 170.65 171.58 172.49 173.37 174.23 175.05 175.85 176.62 177.36 178.07 178.75 179.40 180.02 180.61 181.17 181.69 182.19 182.65 183.08 183.48 183.84 184.17 184.46 184.72 184.95 185.14 185.30 185.42 185.51 185.55 185.57 185.54 185.48 185.38 185.24 185.07 184.85 184.60 184.30 183.97 183.59 183.17 182.71
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS TABLE 7.22.
T (K)
117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
P (Bubble) (MPa)
{continued)
Liquid p (mol/dm^)
0) (m/s)
22.51 22.22 21.91 21.59 21.25 20.90 20.53 20.14 19.73 19.28 18.79 18.24 17.62 16.86 15.87 14.20
450.49 437.29 423.89 410.23 396.30 382.04 367.40 352.31 336.67 320.36 303.21 285.00 265.37 243.75 219.07 189.12
1.85 1.95 2.05 2.16 2.27 2.38 2.50 2.62 2.74 2.87 3.01 3.14 3.28 3.43 3.58 3.72
TABLE 7.23.
P (Dew) (MPa)
Vapor p (mol/dm^)
(o (m/s)
2.463 2.626 2.801 2.989 3.191 3.410 3.648 3.908 4.193 4.510 4.865 5.270 5.740 6.307 7.034 8.127
182.21 181.66 181.08 180.45 179.78 179.06 178.31 177.52 176.68 175.81 174.91 173.96 172.98 171.93 170.79 169.40
1.70 1.80 1.90 2.01 2.12 2.23 2.35 2.47 2.59 2.73 2.86 3.01 3.15 3.31 3.47 3.65
Air—Isobar at 0.101 325 MPa
T (K)
p (mol/dm^)
100 200 300 400 500 600 700 800 900
0.124 0.061 0.041 0.030 0.024 0.020 0.017 0.015 0.014 0.012 0.011 0.010 0.0094 0.0087 0.0081 0.0076 0.0072 0.0068 0.0064 0.0061 0.0058 0.0055 0.0053 0.0051 0.0049 0.0047 0.0045 0.0044 0.0042 0.0041 0.0039 0.0038
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200
299
CO (m/s)
198.21 283.45 347.36 400.50 446.40 487.07 523.89 557.85 589.60 619.60 648.15 675.47 701.72 727.03 751.50 775.19 798.19 820.54 842.30 863.51 884.21 904.43 924.20 943.56 962.52 981.10 999.34 1017.25 1034.84 1052.13 1069.14 1085.87 {continues)
300
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.23.
(continued)
T (K)
p (mol/dm-^)
CO (m/s)
3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.0037 0.0036 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0030 0.0029 0.0028 0.0028 0.0027 0.0026 0.0026 0.0025 0.0025 0.0024
1102.35 1118.58 1134.57 1150.34 1165.89 1181.23 1196.37 1211.31 1226.07 1240.65 1255.05 1269.29 1283.36 1297.27 1311.03 1324.65 1338.12 1351.45
TABLE 7.24.
Air—Isobar at 1 MPa
T (K)
p (mol/dm-^)
(o (m/s)
100
26.59 25.23 1.38 0.616 0.402 0.300 0.240 0.200 0.171 0.150 0.133 0.120 0.109 0.100 0.092 0.086 0.080 0.075 0.071 0.067 0.063 0.060 0.057 0.055 0.052 0.050 0.048 0.046 0.044 0.043 0.041 0.040
658.25 582.97 185.23 281.74 348.45 402.33 448.46 489.16 525.96 559.86 591.54 621.47 649.95 677.21 703.40 728.66 753.07 776.72 799.67 821.98 843.70 864.87 885.53 905.72 925.46 944.79 963.72 982.28 1000.49 1018.37 1035.94 1053.21
106.22 108.10
200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.24. (continued) T (K) 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
p (mol/dm^) 0.039 0.038 0.036 0.035 0.034 0.033 0.032 0.032 0.031 0.030 0.029 0.029 0.028 0.027 0.027 0.026 0.026 0.025 0.025 0.024
co (m/s) 1070.20 1086.91 1103.37 1119.59 1135.56 1151.31 1166.84 1182.17 1197.29 1212.22 1226.97 1241.53 1255.92 1270.15 1284.21 1298.11 1311.86 1325.46 1338.92 1352.24
TABLE 7.25. Air—Isobar at 5 MPa T (K)
p (mol/dm^)
0) (m/s)
100 200 300 400 500 600 700 800 900
27.22 3.38 2.02 1.49 1.18 0.983 0.843 0.738 0.657 0.592 0.539 0.494 0.457 0.424 0.396 0.372 0.350 0.331 0.314 0.298 0.284 0.271 0.259 0.249 0.239 0.230 0.221 0.213 0.206
710.56 279.42 355.63 411.69 458.30 498.93 535.45 569.01 600.34 629.93 658.09 685.05 710.97 735.96 760.13 783.56 806.30 828.42 849.96 870.96 891.46 911.50 931.10 950.29 969.10 987.54 1005.64 1023.41 1040.88
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900
(continues)
301
302
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.25.
(continued)
T (K)
p (mol/dm-^)
0) (m/s)
3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.199 0.193 0.187 0.181 0.176 0.171 0.166 0.162 0.158 0.154 0.150 0.146 0.143 0.139 0.136 0.133 0.130 0.127 0.125 0.122 0.120
1058.05 1074.95 1091.57 1107.95 1124.08 1139.98 1155.65 1171.12 1186.37 1201.43 1216.30 1230.98 1245.49 1259.82 1273.99 1288.00 1301.85 1315.55 1329.10 1342.52 1355.79
TABLE 7.26.
Air—Isobar at 10 MPa
T (K)
p (mol/dm^)
0) (m/s)
100 200 300 400 500 600 700 800 900
27.86 7.39 4.04 2.92 2.32 1.93 1.65 1.45 1.29 1.17 1.06 0.975 0.902 0.839 0.784 0.736 0.693 0.655 0.621 0.591 0.563 0.538 0.515 0.494 0.474 0.456 0.440 0.424 0.410
763.47 296.30 369.50 425.59 471.81 511.90 547.83 580.83 611.64 640.74 668.46 695.01 720.56 745.21 769.07 792.20 814.68 836.55 857.85 878.64 898.94 918.78 938.21 957.23 975.87 994.17 1012.12 1029.76 1047.10
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900
TABLES OF THE SPEED OF SOUND IN IMPORTANT FLUIDS
TABLE 7.26. (continued) T (K)
p (mol/dm^)
0) (m/s)
3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
0.396 0.384 0.372 0.361 0.350 0.340 0.331 0.322 0.314 0.306 0.298 0.291 0.284 0.277 0.271 0.265 0.260 0.254 0.249 0.244 0.239
1064.15 1080.92 1097.44 1113.71 1129.74 1145.54 1161.12 1176.49 1191.66 1206.63 1221.42 1236.03 1250.46 1264.72 1278.82 1292.76 1306.55 1320.19 1333.68 1347.04 1360.25
TABLE 7.27. Air—Isobar at 50 MPa T (K)
p (mol/dm^)
0) (m/s)
100 200 300 400 500 600 700 800 900
30.94 21.58 15.23 11.68 9.54 8.10 7.06 6.28 5.66 5.15 4.74 4.38 4.08 3.82 3.59 3.38 3.20 3.04 2.89 2.76 2.64 2.53 2.43 2.33 2.24 2.16 2.09 2.02
1012.41 670.59 581.66 582.86 604.06 629.50 655.45 680.98 705.86 730.05 753.58 776.48 798.80 820.55 841.78 862.52 882.80 902.63 922.05 941.07 959.73 978.04 996.01 1013.67 1031.02 1048.10 1064.90 1081.44
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
{continues)
303
304
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS TABLE 7.27.
(continued)
T (K)
p (mol/dm^)
2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
1.95 1.89 1.83 1.78 1.73 1.68 1.64 1.59 1.55 1.51 1.48 1.44 1.41 1.37 1.34 1.32 1.29 1.26 1.23 1.21 1.19 1.16
CO (m/s)
1097.73 1113.79 1129.62 1145.23 1160.63 1175.83 1190.84 1205.66 1220.30 1234.77 1249.06 1263.20 1277.18 1291.01 1304.69 1318.22 1331.62 1344.88 1358.01 1371.01 1383.88 1396.64
T A B L E 7.28.
Air—Isobar at 100 M P a
T (K)
p (mol/dm-^)
0) (m/s)
100 200 300 400 500 600 700 800 900
33.16 26.18 21.14 17.60 15.09 13.23 11.80 10.67 9.75 8.98 8.33 7.77 7.29 6.86 6.48 6.15 5.85 5.57 5.32 5.10 4.89 4.70 4.52 4.36 4.21 4.06 3.93 3.81
1192.37 932.01 818.47 779.80 772.41 778.24 790.14 805.13 821.78 839.35 857.39 875.65 893.95 912.18 930.28 948.19 965.90 983.39 1000.64 1017.65 1034.43 1050.98 1067.31 1083.41 1099.29 1114.97 1130.44 1145.72
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
REFERENCES
305
TABLE 7.28. (continued) T (K)
p (mol/dm^)
0) (m/s)
2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000
3.69 3.58 3.48 3.38 3.29 3.20 3.12 3.04 2.97 2.90 2.83 2.76 2.70 2.64 2.59 2.53 2.48 2.43 2.38 2.34 2.29 2.25
1160.81 1175.72 1190.45 1205.01 1219.40 1233.63 1247.71 1261.63 1275.41 1289.04 1302.54 1315.90 1329.13 1342.24 1355.21 1368.07 1380.81 1393.43 1405.95 1418.35 1430.64 1442.83
Additional Reading Kinsler, L.E., Frey, A.R., Coppens, A.B., and Sanders, J.V. (1982). Fundamentals of Acoustics New York: John Wiley & Sons. Thompson, P.A. Compressible-Fluid Dynamics New York: McGraw Hill. (1972). Trusler, J.P.M. (1991). Physical Acoustics and Metrology of Fluids Bristol: Adam Hilger. Van Dael, W. (1975). In Experimental Thermodynamics of Non-Reacting Systems (International Union of Pure and Applied Chemistry, Commission on Thermodynamics and Thermochemistry; Le Neindre, B. and Vodar, B. eds. London: Butterworths. chapter 7, Thermodynamic Properties and the Velocity of Sound, pp. 527-575.
References 1. Slattery, J.C. (1981). Momentum, Energy, and Mass Transfer in Continua, New York: Krieger Pubhshing. 2. Thompson, P.A. (1972). Compressible-Fluid Dynamics New York: McGraw Hill. 3. NIST Thermophysical Properties of Pure Fluids Database (NIST12, Version 5.0). (2000). Natl. Inst. Stand. Tech., Gaithersburg, MD. 4. McQuarrie, D.A. (1976). Statistical Mechanics New York: Harper & Row.
306
SPEED OF SOUND AS A THERMODYNAMIC PROPERTY OF FLUIDS
5. Costa Gomes, M.F., and Trusler, J.P.M. (1998). The speed of sound in nitrogen at temperatures between T = 250 K and T = 350 K and at pressures up to 30 MPa, J. Chem. Thermodynamics 30: 527. 6. Trusler J.P.M. (1991). Physical Acoustics and Metrology of Fluids Bristol: Adam Hilger. 7. Van Dael, W. In Experimental Thermodynamics of Non-Reacting Systems (International Union of Pure and Applied Chemistry, Commission on Thermodynamics and Thermochemistry Le Neindre, B. and Vodar, B, eds. London: Butterworths, chapter 11: Thermodynamic properties and the velocity of sound, pp. 527-575. 8. Reid, R.C., Prausnitz, J.M., and Poling, B.E. (1987). The Properties of Gases and Liquids New York: McGraw Hill. 9. Camahan, N.F., and Starling, K.E. (1969). Equation of state for nonattracting rigid spheres, J. Chem. Phys. 51: 635. 10. Younglove, B.A. (1982). Thermophysical properties of fluids. I. argon, ethylene, parahydrogen, nitrogen, nitrogen trifluoride, and oxygen, J. Phys. Chem. Ref Data 11, pp. 1-1 to 1-353 (368 pages). 11. Sengers, J.V., and Levelt Sengers, J.M.H. (1978). In Progress in Liquid Physics (Croxton, C.A., ed., Chichester, U.K: Wiley. Chapter 4: Critical phenomena in classical fluids, pp. 103-174. 12. Tegeler, Ch., Span, R., and Wagner, W. (1999). A new equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressures up to 1000 MPa, J. Phys. Chem. Ref Data, 28: 779. 13. Estrada-Alexanders, A.F., and Trusler, J.P.M. (1995). The speed of sound in gaseous argon at temperatures between 110 K and 450 K and at pressures up to 19 MPa. J. Chem. Thermodyn. 27: 1075. 14. Span, R., Lemmon, E.W., Jacobsen, R.T, and Wagner, W. (1998). A reference quality equation of state for nitrogen. Int. J. Thermophys. 19: 1121. 15. lAWPS. (1995). Release on the Formulation for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use; copies of this and other lAPWS releases can be obtained from the Executive Secretary of lAPWS, Dr. R.B. Dooley, Electric Power Research Institute, 3412 Hill view Ave., Palo Alto, CA 94304 or at http://www.iapws.org. 16. Lemmon, E.W., Jacobsen, R.T, Penoncello, S.G., and Friend, D.G. (2000). Thermodynamic properties of air and mixtures of nitrogen, argon, and oxygen from 60 to 2000 K at pressures to 2000 MPa. J. Phys. Chem. Ref Data 29: 331. 17. Younglove, B.A., Frederick, N.V., and McCarty, R.D. (1993). Speed of Sound Data and Related Models for Mixtures of Natural Gas Components, NIST Monograph 178. 18. Lemmon, E.W., and Jacobsen, R.T. (1999). A Generalized Model for the Themodynamic Properties of Mixtures, Int. J. Thermophys. 20: 825. 19. Thermodynamics Research Center; Texas Engineering Experiment Station, College Station, TX (1961-2000): Natl. Inst. Stand. Tech., Boulder, CO (2000-).