Compressibility equations for liquids: a comparative study

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BRIT. J. APPL. PHYS., 1967, VOL. 18. PRINTED IN GREAT BRITAIN

Compressibility equations for liquids : arative study

A. T. J. HAYWARD Fluids Group, Properties of Fluids and Basic Heat Transfer Division, National Engineering Laboratory, East Kilbride, Glasgow

MS. received 5th December 1966, in revised fovm 20th Februavy 1967

Abstract. Various empirical equations used for expressing the compressive properties of liquids have been studied. Despite its popularity, the well-known equation attributed to Tait has several undesirable features; moreover, it is in fact not Tait’s original equation, but is the result of an accidental misquotation by a later writer.

Tait’s original equation, when rearranged, leads to an equation of the form

____ - K~ + mP vo - v where KO is the bulk modulus at zero pressure and m the slope of the bulk-modulus- pressure curve.

The equations of Tumlirz and of Tammann are merely rearrangements of the above equation. The spurious version of the Tait equation, Hudleston’s equation, MacDonald’s equation and the Van der Waal equation of state are all asymptotic to it at zero pressure, and are practically equivalent to it over the normal range of application.

The above equation is both the most accurate and the most convenient two-constant compressibility equation available. By the addition of a term in P2 the equation can be adapted to fit data for water up to 12 kb; a further term in P3 is needed to accom- modate a wide range of organic liquids up to 12 kb.

1. Introduction The subject of compressibility equations for liquids is really a very simple one, Un-

fortunately, it has been so badly treated in the literature that it has been made to appear unnecessarily complex, and the resulting confusion has had at least one serious consequence. Two generations of workers have been misled into using what they have come to call ‘Tait’s equation’, without realizing that this well-known equation is not Tait’s original equation, and that another equation closer to Tait‘s original is available which is simpler, more convenient to use and fits experimental data at least as well as (or, in some cases, much better than) the so-called Tait equation.

The purpose of this paper is therefore threefold: to clarify an unnecessarily obscure subject, to show the advantages of using the modified form of the true Tait equation instead of the popular equation wrongly attributed to Tait and to discuss the use of compressibility equations at high pressures, where all the existing two-constant equations fail to express experimental data adequately.

2. Basic principles Many attempts have been made to derive a compressibility equation from molecular

theory, but none of them has resulted in a convenient equation expressing the results of experiments with adequate accuracy. To meet this need it is necessary to employ some empirical equation, the sole justification for which is that it works.

The construction of such an equation is a relatively simple matter, on account of the shape of the compression curve of a typical organic liquid, which is illustrated in figure 1. When the applied pressure is plotted against the consequent volume change, for either an isothermal or an isentropic pressure change, the result is a smooth curve of relatively small

965

966 A . T. J. Hayward

curvature. Because of this small curvature the curve over the first few tens of bars may, for all practical purposes, be regarded as a straight line. Within this range a liquid may be regarded as obeying Hooke’s law, and only one constant is needed to define its com- pressive properties, namely the bulk modulus of elasticity, which is equal to the (constant) slope of the compression curve.

Within the range of hundreds of bars the departure from linearity of the compression curve can no longer be neglected, but it is still so small that the curve can be represented adequately by a quadratic passing through the origin. It is not until pressures of the order of kilobars are encountered that the departure from linearity becomes sufficiently great to necessitate the use of a cubic equation.

Reduction in volume CV,-V)./V,

Figure 1. Compression curve for a liquid. Ranges and equations applicable: A, tens of bars, linear; B, hundreds of bars, quadratic; C, kilobar range, cubic.

In practice the use of two-constant compressibility equations is therefore confined to pressures below 1 kb with organic liquids. With water, aqueous solutions and liquid metals, however, the curvature of the compression curve is such that it can still be repre- sented by a quadratic at pressures up to several kilobars.

The quadratic selected may be either with P as a function of h V or with A V as a function of P. It is obviously easier in practice, however, to derive a linear relationship between dVjdP (or its reciprocal) and either P or V than to derive the corresponding quadratic relation between P and AV. Further, since the P,‘ilV curve is concave towards the pressure axis, as P tends towards infinity dPjdV also tends towards infinity while dVjdP tends towards zero, and consequently a plot of dPjdV against P is inherently more likely to give a linear fit.

This simple fact is the basis for a whole family of two-constant compressibility equations, which may be called ‘linear bulk-modulus equations’.

3. Linear bulk-modulus equations

its reciprocal, which is called bulk modulus, is defined by Compressibility is defined as fractional volume change per unit change in pressure, and

dP K G - V - dV’

This is the only true definition of bulk modulus, but to distinguish the quantity so defined from certain broadly similar quantities it is often referred to as the ‘tangent bulk modulus’. To derive the value of K at P = PI from a series of measured values of P and Vit is necessary first to plot P against VO - V, and then to derive the slope of the curve at the point (PI, VI) and finally to multiply this slope by VI. If the measurement of the slope is done graphically the result is likely to be rather inaccurate, and it is usually necessary to use a numerical process, which involves either a considerable expenditure of effort or the use of a computer.

967 Compressibility equations for liquids: a conzparatiue study

It is very much easier to derive another quantity, which may be regarded as an average value of bulk modulus over the range from 0 to P. It is defined by

- VOP K E - vo - Y’

Because it corresponds to the slope of the secant (or chord) cutting the compression curve at the origin and at the point (P, V), this quantity is usually referred to as the ‘secant bulk modulus’.

A third quantity has also been introduced to which the term ‘bulk modulus’ is sometimes applied. This quantity is a hybrid between the two quantities defined above as K and R, and is defined by

dP K/e-vo- dV‘ This quantity has no commonly accepted distinctive name, but it can fittingly be termed

the ‘mixed bulk modulus’. It serves no useful purpose, since it is neither applicable to thermodynamic analysis nor convenient for engineering calculations, and it seems a pity that it was ever invented.f

All three forms, K, R and K‘, coincide when P is zero, but diverge at an increasing rate as P increases.

At pressures up to several hundred bars with any liquid, K, R and K’ are all practically linear functions of P. If the results of a compressibility experiment within this range are expressed as a plot of either K, R or K‘ against P, the deviation of the experimental points from a straight line is likely to be random, and ascribable to experimental error in the apparatus used. This means that any one of the following three equations provides an adequate way of expressing experimental results within this pressure range, so it is not possible to distinguish between them on grounds of accuracy.

(a) The linear tangent-modulus equation (used by Moelwyn-Hughes (1957) and Anderson (1 966)) :

(4) dP dV K = - V--Ko-mlP.

(b) The linear secant-modulus equation (used by Klaus and O’Brien (1964), Holland (1966) and Hayward (1964)):

(e) The so-called Tait equation (used by a very large number of authors), which is identical with the linear mixed-modulus equation :

(6) dP dV K’ E - YO - = KO + nzsP.

It will be observed that whereas KO, the value of bulk modulus at zero pressure, is the same in all three equations, the slope m of the bulk-modulus-pressure line is different in each equation.

Since all three equations are equally accurate within the pressure range that concerns the great majority of workers, the choice between them can be governed by convenience. On this ground the linear secant-modulus equation (5) is unquestionably superior because R can very easily be evaluated from the experimental data. The use of this equation is also in harmony with the philosophical principle of Occam’s razo?, which insists that the simplest possible hypothesis should always be adopted first, and abandoned for a more

- P. Fortunately no writer so far appears to have confused the issue still further by using this form Qf bulk modulus, and so its hypothetical existence can be ignored.

f The other hybrid, K“ 3 VP/( VO - V ) is, of course, also possible. It is equivalent to

968 A. T. J. Hayward

complex one only when it is shown to be inadequate. Where an algebraic equation is adequate, there is no point in using a differential equation.

It is therefore most surprising to find that there has been an almost universal preference for the so-called Tait equation (6). This and (4) are both very inconvenient to use in practice because both K and K‘ involve a differential coefficient which cannot easily be evaluated from experimental readings. Even if, as is customary, the equations are integrated, they then involve logarithmic functions which are equally inconvenient to fit to the results of experiments.

4. The linear secant-modulus equation Klaus and O‘Brien (1964), Holland (1966) and Hayward (1964) have independently

been using the linear secant bulk-modulus equation for several years. The primary reason for the choice of this equation has already been given: its great simplicity of application. A second valuable feature of this equation is that its two constants have an easily grasped physical significance. For technological purposes these two features provide ample justification for preferring this equation.

I 201 ’ ’ 0 0.2 0.4 0.6 0.8

P ( k b l Figure 2. Linear relationship between and P. (Results of Kell and Whalley (1965) for water

at IO’c and 0-i kb.)

There is, however, a third reason for preferring this equation, in that it is the only com- pressibility equation which can be used to express the result of tests on water at pressures up to several kilobars. Figure 2 shows the linear relationship between R and P given by the results of some recent measurements up to 1 kb by Kell and Whalley (1965), who took elaborate precautions to obtain the highest possible accuracy. The most accurate values available for the range 0-3 kb are generally conceded to be those of Amagat (1893), which also give a linear relationship between R and P, as is shown in figure 3. Although the results at one temperature only are given in each case, similar graphs have been drawn for tests at other temperatures and similar linear relationships were always obtained.

In both figures 2 and 3 there is a small departure from linearity at very low pressures, but that this is without physical significance may be deduced from the fact that at the lowest pressures the Kell and Whalley points deviate upwards, while those of Amagat deviate downwards, In both cases this deviation at low pressures is almost certainly due to the increased proportional error which is inevitable when very small values of volume change have to be measured.

Only when a compression test on water is extended to about 10 ltb does the curvature of the ($P) graph become clearly apparent. Bridgman (1958) worked over this range, and although his results when expressed in this form show a departure from linearity, this is still less than 2 % of the mean value. This is not very much greater than the probable error in the data.

The superiority of the linear secant-modulus equation (5) can be seen by comparing figure 3 with figure 4, in which comparable values of K and K’ derived from Amagat’s results have been plotted against P. The departure from linearity amounts to about 1 * 5 %

Compressibility equations for liquids: a comparatire study 969

201 I , 0 I 2 3

P ( k b l P ( k b l

Figure 3. Linear relationship between and Figure 4. Nonlinear relations between K P. (Amagat's (1893) results for water at and P and between K' and P. (Amagat's

(1893) results for water at 2 0 . 4 " ~ and 0-3 kb.) If the so-called Tait equation held for water between 0 and 3 kb, the K points would

fall on the straight line drawn here.

2 0 . 4 " ~ and 0-3 kb.)

of the mean value in the K graph and about 2 % in the K' graph, thus showing that neither (4) nor (6) is valid for water over the range 0-3 kb.

Yet another important advantage of the linear secant-modulus equation is that, because of its simple form, it can be used to derive several important parameters, such as the specific volume and density at any pressure P and the true or 'tangent' bulk modulus K.

Whatever the relationship between K and P, it follows from (2) that

x - P v= Vo- K and hence that

poR p=-- K - P

where p and po are the densities at the pressures P and Po. It also follows from (2) that, whatever the relationship between i? and P,

K = R(R - P ) R - P dR]dP

and

(7 )

(9)

For the special case of the linear secant-modulus equation, which may be expressed in

= KO - J??P the form

it is additionally true that (1 1)

R KO

K = -(I?- P )

and R2

K' = -, KO

970 A. T. J . HayM.ard

5. The original Tait equation In 1888 Tait proposed his compressibility equation. He re-stated this equation in

several subsequent papers (Tait 1900), but there is no record of his ever having proposed any other compressibility equation. Present-day users of the equation to which his name is nowadays attached may find it hard to believe that this is not the equation that Tait proposed, and the relevant passage from his first paper will therefore be quoted in full:

"Thus the average compressibility through any range of pressure falls off more and more slowly as that range is greater. And, within the limits of my experiments, I found that this relation between pressure and average compressibility could be fairly well represented by a portion of a rectangular hyperbola, with asymptotes coincident with and perpendicular to the axis of pressure. Hence at any one temperature (within the range I was enabled to work in), if CO be the volume of fresh water at one atmosphere, 2; that under an additional pressure p , we have

very nearly, A and IT being quantities to be found." It is clear from the left-hand side of (14) that the variable designated by Tait as 'average

compressibility' is the reciprocal of the secant bulk modulus. If his equation is inverted it becomes

This equation is of identical form with (ll), but with II/A = KO and A-1 = m. Thus the equation propounded by Tait was actually the linear secant-modulus equation, expressed in reciprocal form.

Why Tait chose to express it in this form, and then laboriously to fit his experimental points to a hyperbola, when he could have inverted the equation and fitted his points to a straight line, will always remain a mystery. All that can now be said is that if only he had done so, he might have saved himself from being misquoted and two generations of physicists from being led astray in consequence.

6. The spurious Tait equation It will never be known for certain who it was that first misquoted Tait. The earliest

occurrence of the spurious Tait equation discovered by the author, however, is in a book published by Tammann (2907). But Dr. G. S. Kell (who independently reached the same conclusion as the present author) has said in a private communication that Tammann (1895) actually attributed the wrong equation to Tait as early as that year, in a paper published during Tait's lifetime.

Whether Tammann himself misquoted Tait, or whether he was merely following some unknown earlier writer who had misquoted Tait, is of little consequence. The fact is that hundreds of authorities during the past sixty years have blindly followed Tammann and used this inferior equation instead of the true Tait equation, so chat Tammann probably deserves to be regarded as the perpetrator of one of the most far-reaching misquotations in the history of physics.

The equation which he attributed to Tait was, in the notation of the present paper.

dV A dP B - P' -

Comparison with (14) reveals that Tammann has replaced Tait's 'average compressi- bility'

' O - ' (which could be written as - - PVO vo AP

Compressibility equations for liquids: a comparative study 971

by the corresponding differential coefficient, dV/dP. The resulting equation is the linear mixed-modulus equation (6), as may be seen by inverting both sides of (16) and multiplying throughout by the constant VO.

Tammann then proceeded to integrate this equation, obtaining

It was left to later writers to convert this integrated equation into the form in which it is generally used today:

vo - v B + P -- - Clog (B). VO

It cannot be emphasized too strongly that this equation has no advantages over the linear secant-modulus equation, and has a number of serious disadvantages. It is much less convenient to use, its constants have no obvious physical meaning, it does not give rise to simple expressions from which other important constants can be obtained and it does not fit experimental data for water nearly as well as the other equation, as may be seen from a comparison of figures 3 and 4.

This last statement is, of course, in conflict with the views of various modern writers who have extolled the virtues of the spurious Tait equation. Some of them (see for example Rowlinson 1959, p. 32) have quoted the following table from Tait's papers as an alleged proof of the accuracy of the spurious Tait equation.

Pressure (atm) 1 501 1001 1501 2001 2501 3001 Relative volume of water 1 .OOOOO 0,97668 0.95645 0.93924 0,92393 0.91065 0.89869

at 0"c observed by Amagat

at 0"c calculated by Tait

Reference to Tait's original paper (Tait 1900, pp. 334-8) reveals, of course, that he based his calculations upon his own equation (equation (14)), so that t h s table is really a con- firmation of the linear secant-modulus equation, and not of the spurious Tait equation.

Relative volume of water 1-00000 0.97657 0.95652 0.93916 0.92399 0.91062 0.89875

7. Equivalence of the constants in the spurious Tait equation Although there is no reason why the spurious Tait equation should continue in use in

the future, a great deal of work in the past has been published in terms of the two constants in this equation. It is therefore fortunate that these two constants can easily be converted to the two constants in the linear secant-modulus equation.

When the spurious Tait equation is expressed in the usual form (equation (18)) the cor- responding differential form is

dP B 2.3026p. dV C C

K' E - Vo- = 2.3026 - $- - (19)

At zero pressure K' = X = KO, and consequently it follows from (19) that

Expressing m in terms of B and C is not quite so simple, since (11) and (19) are incom- patible. That is to say, if K' is a linear function of P, then Rcannot also be linear (although it is very nearly linear when P < K). Equation (13), however, although strictly true only when R is linear, is a very close approximation in the case of (19), provided that P <I?. From this it follows that for a (K, P ) curve corresponding to a set of points satisfying equation (17)

dl? _ - - 1 E 1 3 ( ~ B j l i 2 (21) d p B - P '

9 72 A . T. J. Hapsaid

This set of points will form a very flat curve. and if the best straight line is drawn through them its slope will be practically the same as the value of dK/dP at the mean pressure. Hence the required relationship is

where Pmas is the upper extreme of the pressure range over which the experiments were carried out. (It is assumed that the lower extreme is at or near atmospheric pressure.)

Over fairly small pressure ranges the factor in parentheses can be neglected, with a consequent simplification of (22). Even over the greatest pressure ranges for which the spurious Tait equation can be expected to hold, the factor in parentheses will have only a small effect on the value of 112, which remains nearly proportional to the reciprocal of C. Consequently, the observation which has frequently been made about C in the spurious Tait equation-that it is approximately the same for nearly all organic liquids at any temperature-applies also to the constant in in the linear secant-modulus equation.

8. The Tumlirz and the Tammann equations Tumlirz (1909) proposed the equation

where p and t' denote pressure and volume, P and a are arbitrary constants and the right- hand side of the equation is (for isothermal compression) another constant. This is expressed in the present notation as

(P - A)( V - B ) = C. (23) Tammann (191 1) proposed the equation

where V, is the volume when P = cc and A and K are arbitrary constants. It is typical of the present confused thinking in this field that these are still regarded as

two distinct equations (Partington 1951), although they are nothing of the kind. By introducing the term VO, the volume at zero pressure, and using it to eliminate one of the other constants, the equations can be rearranged thus :

Tumlirz :

Tammann

Equations (25) and (26) are both clearly of the linear secant-modulus form, thus showing that both Tumlirz's and Tammann's equations are nothing more than rearrangements of the linear secant-modulus equation.

9. The Hodlestm and tfie M~~D~plaliZ equations There is almost endless scope for constructing complicated compressibility equations,

which work merely because they virtually coincide with the linear secant-modulus equation at relatively low pressures. The equation proposed by Hudleston (1937), and subsequently advocated by Bett (i953), and that proposed by MacDonald (1966) are examples of this.

Hudleston's equation is of the form

(27)

973 Compressibility equations for liquids: a coniparatice study

where L = L o ( V / V O ) ~ / ~ and Lo, C1 and C2 are arbitrary constants. Substituting for L and rearranging gives

where C3 and C4 are arbitrary constants. MacDonald's equation is of the form

where KO is the bulk modulus at zero pressure and n is an arbitrary constant. It can be shown that both these equations are asymptotic to the linear secant-modulus

equation at zero pressure and, within the limits of experimental error, do not differ from it significantly over the normal range of application (0-1 kb for organic liquids). There is therefore nothing to be gained by using such equations as these instead of the simpler and more convenient linear secant-modulus equation.

10. Extension to higher pressures There is little demand for information on the compressibility of liquids at pressures

above 1 kb, and there appears to be only two sources of data at pressures up to 10 kb and beyond, namely the works of Bridgman (1964) and the A.S.M.E. Pressure-Viscosity Report (1953). Between them these cover a large number of liquids at temperatures from 0 to 220°c, and it is of interest to see what form of equation could be used to cover this wide range of pressure. An analysis of these results has shown that, for most liquids, a polynomial of the fourth degree is required to express the relationship between pressure and volume change. It is therefore evident that, in general, no two-constant equations could be expected to provide a reasonable fit.

Ideally, a compressibility equation for use up to very high pressures should fulfil the foilowing requirements :

(a) It must fit any liquid over the full range of interest in pressure and temperature, with an accuracy commensurate with the accuracy of the experimental data.

(b) It must be convenient to use. (c) It must coincide with the linear secant-modulus equation at low pressures. (d) It must employ the minimum number of arbitrary constants. Requirements (b) and (c) point clearly to the desirability of expressing I? as a polynomial

in P. Requirement (d) indicates that this polynomial should be of as low a degree as possible, consonant with requirement (a). Large numbers of graphs of K against P have been plotted from the data given in Bridgmau (1964) and the A.S.M.E. Report (1953), and from an examination of these it is clear that the degree of the polynomial necessary is different for water, mercury and organic liquids.

With water the slope of the (I?, P) graph does not change very much, and the rate of change is practically constant. Consequently a second-degree relationship between K and P provides an adequate fit. This is illustrated in figure 5, where Bridgman's results at 5 0 " ~ are given as typical of the results obtained with water. A second-degree curve has been drawn through the points by the method of least squares, and provides an excellent fit.

All the existing data for water within this pressure range at various temperatures can be fitted by equations of the form

where KO, m and n are all positive constants. The only results available for mercury at pressures up to 12 kb are those obtained by

Bridgman at 20"c. These are plotted in figure 5 . To fit the points with high precision a second-degree polynomial must be used, similar to equation (30) but with the sign of the last term reversed because the curve for mercury would have to be concave upwards.

K = KO -+ mP - nP2 (30)

974 A . T. J. Hayward

Such a curve, however, has not been drawn through the points because, for several reasons, it seems unlikely to be justified.

In the first place, the graphs for every other liquid that has been examined are curved in the opposite direction, and there is no reason to expect that mercury would prove exceptional in this respect. Secondly, it is physically impossible for the upward curvature to continue very far, since it is bound to lead to an infinite value of K (though not of K) at a finite pressure; a second-degree polynomial drawn though the points for mercury in figure 5 would lead to an infinite value of K at about 40 kb. Thirdly, Bridgman criticized his own experiments with mercury, on the grounds that his piezometers appeared to be deforming anisotropically under pressure. This caused his results to be widely scattered, and he employed an elaborate smoothing technique to deal with the scatter. The upward curvature in figure 5 is probably a direct consequence of Bridgman’s smoothing technique.

Figure 5. Secant bulk modulus of four liquids at pressures up to 12 kb. Curve A, water (50”c, quadratic); €3, di(2-ethylhexyljsebacate (25 ‘c, cubic) ; C, iso-amyl alcohol (20”c, cubic); D, mercury

(20”c, linear).

Consequently a straight line has been drawn through the points. The line is within 1 % of every point, and this is considerably less than the probable error of the data. In the present rather inadequate state of knowledge, therefore, it is evident that the linear secant-modulus equation (equation (1 1)) is capable of expressing the compressive properties of mercury up to at least 12 kb.

Although with water, as has already been mentioned, the value of dzK/dP* is practically constant, with all organic liquids it appears to decrease with increasing pressure. Two typical sets of data for organic liquids, one from each source, are given in figure 5, in which the increased curvature at the lower pressures can be clearly seen. Because the rate of decrease of d2K/dP2 may, within the limits of accuracy of the available data, be regarded as constant, a third-degree polynomial can be used to express the compressive properties of any organic liquid up to 12 kb. This will take the form

K = KO + w P - I z P ~ + ~ P ~ (31)

where KO, m, n and q are all positive constants. A further advantage of expressing the compressive properties of liquids by an equation

where Kis given as a polynomial in P is that K, and hence its reciprocal, the compressibility, can easily be derived from such an equation.

With the quadratic secant-modulus equation, for use with water up to very high pressures,

Compressibility equations for liquids: a comparatice study

it follows from (9) that when

then R(R - P) K = KO nP2'

975

Similarly, with the cubic secant-modulus equation, for use with organic liquids up to very high pressures, it follows from (9) that when

then R(R - P)

KO -+ nP2 - 2qP3' K = (33)

11. Van der Waal's equation Although the Van der Waal equation of state is not a compressibility equation in the

usually accepted sense, it is of interest to compare it with the other equations because of its theoretical significance. For isothermal conditions the equation is

This can be rearranged to give

Both the expressions within parentheses in (35) are constants, and since V changes only very slowly with P the factor Vo2/V2 can be ignored at low pressures. Consequently this equation also coincides with the linear secant-modulus equation at zero pressure. At higher pressures, however, the factor Vo2/ V 2 becomes increasingly significant, and the equation gives much higher values of i? at high pressures than are given by the linear secant-modulus equation-while real liquids at high pressures deviate from the linear secant- modulus equation in the opposite direction.

12. Conclusions (a) Several two-constant empirical equations may be used to express the compressive

properties of liquids over the most commonly used pressure range, from zero to several hundred bars. Of these, the linear secant-modulus equation is for severzl reasons the best. By the addition of one or two extra terms it can be extended to cover all liquids at pressures up to 12 kb over a wide range of temperature (see $13).

(b) The well-known equation to which Tait's name has been attached is not a desirable equation, and was certainly not propounded by Tait. It appears to have originated through an unfortunate misquotation by Tammann of Tait's original equation (which was actually the linear secant-modulus equation expressed in reciprocal form). Relationships have been derived between the so-called Tait constants and the constants in the linear secant-modulus equation, thus enabling results expressed in one form to be re-expressed in the other (see $13).

(e) The equations proposed by Tumlirz and by Tammann both fit experimental data well, but only because they are both identical with the linear secant-modulus equation, which these writers have expressed in less convenient forms. The so-called Tait equation, and the equations proposed by Hudleston and by MacDonald, are all asymptotic to the linear secant-modulus equation at zero pressure and, within the limits of experimental error, may be regarded as coinciding with it over the normal range of application (0-1 kb); these three equations, however, are much less convenient to use than the linear secant- modulus equation. Van der Wad's equation of state is also asymptotic to the linear

9 76 A . T. J. Hayward secant-modulus equation at zero pressure, but does not fit experimental data at all well at moderately high pressures. No two-constant equation is satisfactory at very high pressures.

13. Summary of useful equations 13 . 1. DeJinitiom and relatioriships folloir.ingfr.oiii them

Secant bulk modulus : - YOP K r - vo - V '

VdP K = - - d V '

K - P Y= Yo- K

K - P dRldP' K =

Tangent bulk modulus:

K(K - P)

13 .2. The linear secant-modulus equation This is the best empirical equation for expressing the results of coinpressibility experi-

ments on organic liquids up to pressures of about 0.5 or perhaps 1 kb, on water up to about 3 kb and on mercury up to at least 12 kb.

from which it follows that K = KO - WIP

K = X(K - P ) KO '

13 .3 . The quadratic recarit-modulus equation This is the best equation for water over very large pressure ranges.

X = KO - UTP - nP2 from which it follows that

K(K - P) K = KO nP2 *

13 .4. The cubic secarzt-inoddus equation This is the best equation for organic liquids over very large pressure ranges.

K = KO - nzP - 11P2 - qP3

(where KO, m, 72 and q are all positive constants), from which it follows that K(K - P ) K = KO - nP2 - 2qP3'

13.5. The so-called Tait equation Because this equation is in several ways inferior to the linear secant-modulus equation,

there is no reason why it should continue to be used. Where the results of previous work have been expressed in terms of the constants B and C in the equation

they can be re-expressed in terms of the constants KO and in in the linear secant-modulus

Compressibility equations for liquids: a comparative study 977 equation through the relationships

B KO = 2.3026 - C

and

where Pm,, is the upper extreme of the pressure range over which the experiments were carried out, the lower extreme being assumed to be at or near zero.

Acknowledgments The painstaking work of Mrs. A. C. Findlay in fitting a very large volume of experimental

data to various compressibility equations is gratefully acknowledged. This paper is published by permission of the Director of the National Engineering

Laboratory of the Ministry of Technology and the Controller of Her Majesty’s Stationery Office.

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