A derivation is presented that corrects an expression for the effective acoustic nonlinearity parameter of a mixture of immiscible liquids. The derivation is based upon a mass fraction, rather than volume fraction, formulation

Apfel, R. E.

Chu, B. T.

Everbach, E. Carr

Jiang, P.

Zhu, Z.

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Swarthmore CollegeWorksEngineering Faculty Works Engineering1-1-1991A Corrected Mixture Law For B/AE. Carr EverbachSwarthmore College, ceverba1@swarthmore.eduZ. ZhuP. JiangB. T. ChuR. E. ApfelFollow this and additional works at: http://works.swarthmore.edu/fac-engineeringPart of the Engineering CommonsThis Article is brought to you for free and open access by the Engineering at Works. It has been accepted for inclusion in Engineering Faculty Works byan authorized administrator of Works. For more information, please contact myworks@swarthmore.edu.Recommended CitationE. Carr Everbach, Z. Zhu, P. Jiang, B. T. Chu, and R. E. Apfel. (1991). "A Corrected Mixture Law For B/A". Journal Of The AcousticalSociety Of America. Volume 89, Issue 1. 446-447.http://works.swarthmore.edu/fac-engineering/86A corrected mixture law for B/A E. Carr Everbach, a)Zhe-ming Zhu, b) Peng Jiang, BoaTeh Chu, and Robert E. Apfel Yale Center for Acoustics, Yale University, 2].59 Yale Station, New Haven, Connecticut 06520 (Received 7 December 1989; revised 30 August 1990; accepted 17 September 1990) A derivation is presented that corrects an expression for the effective acoustic nonlinearity parameter of a mixture of immiscible liquids. The derivation is based upon a mass fraction, rather than volume fraction, formulation. PACS numbers: 43.25.Ba Recent interest in applying measurements of the acous- tic nonlinearity parameter B/A of biological materials to in- fer tissue composition•'2 has led to a re-examination 3 of the underlying mixture laws for B/A. In his 1983 paper, 4Apfel derived a relation that gives the effective nonlinear param- eter of a system of immiscible liquids, given the densities, compressibilities, B/A values, and volume fractions of the components. For a system of n components, he showed that /3= •/3,X, (1) i=1 and /•2 B= Xi• • i (2) where/3• and X• are the compressibility and volume fraction of component i, respectively, and the unsubscripted values refer to effective properties of the mixture. Equation ( 1 ) is just the familiar mixture law for compressibility that Cham- br(• has shown can be derived directly from considerations of conservation of mass. Equation (2) was derived by con- sidering the effect of applying a pressure increment to the mixture, but its derivation fails to consider that the relative volume fractions themselves may change due to the applica- tion of the pressure increment. Although the contribution of this change in volume fraction with pressure may be taken into account explicitly, 6we present an alternative derivation that avoids this difficulty by formulating the problem in terms of mass fractions. Let us define Y• as the mass fraction of the ith compo- nent of an n-component mixture: Yi = m•/mtot , ( 3 ) where Mi is the mass of component i and Mtot is the total mass of the mixture. Note that Yi and Xi are related to one another by X i = (p/p•) Y•, where p is the density of the mixture and p• is the density of the ith component. Let us now consider the intensive quantity v = l/p, the specific volume (volume per unit mass ) of an n-component mixture. a)Current address: Department of Engineering, Swarthmore College, Swarthmore, PA 19081-1397. Current address: Institute of Acoustics, Nanjing University, The People's Republic of China. If we assume that the total volume of the mixture equals the sum of the component volumes (i.e., we assume the compo- nents are not interactive and are not mutually soluble), it follows that If Eq. (4) holds, so do its partial derivatives with respect o pressure p (the entropy s is held constant in all differentia- tions). Note that while the component volume fractions of a mixture may change with pressure, the mass fractions do not: 3Yi/•p = 0. The result of differentiating equation (4) is, therefore, 8v = • \8p/s Y,, (5) s i=1 and likewise to higher orders, •p2 S i=l k •p2/s ri. (6) Substituting p = 1/v into the left-hand side of Eq. (5) and noting that the sound speed c2 = (•p/•p)s yields op s- =- .(7) Since Eq. (7) must be true for the mixture and also for each component indioMually, Eq. (5) becomes p2C2 2 2 i= 1 piCi Making use of the expressions /3= (pC2) -• and Xi = (P/Pi) Yi results in an expression relating the compress- ibility of the mixture to that of the components, viz., • = • io Yi •'•-i = • Xi•i , (8) i=1 tOi i=1 which is just Eq. ( 1 ). Similarly, we can use Eq. (6) to arrive at a mixture law for B/.,4. Using the relation B/.,4 = 2pc(8c/8œ)s, the left- hand side of Eq. (6) can be written as: 446 J. Acoust. Sec. Am. 89 (1), January 1991 0001-4966 / 91 / 010446-02500.80 ¸ 1990 Acoustical Society of America 446  Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP:  130.58.65.20 On: Thu, 21 May 2015 17:02:47c2p•' s = ' c2p • p•c 2 -•p •q-p•'c  -•p • p3cn 1 + pc --p3c• 1+-•- . (9) Since Eq. (9) must be true for both the mixture and its com- ponents, Eq. (6) becomes p3cn 1 -Jr- = 3 4 i= ! piCi where all quantities on the left-hand side are properties of the mixture as a whole. Using/3 = (pc 2) -• and the definition K--= 1 q- B/2A results in the desired relation i---1 Pi i=1 (lO) This expression is similar to Eq. (2), but relates the B/A value of the mixture to those of the components through K, the coefficient of nonlinearity. Actually, Apfel first report- ed 7 the result given in Eq. (10), but then later incorrectly modified it, substituting B/A for K. The coefficient of non- linearity K contains information about nonlinearities arising from convection, 8 as well as those from the equation of state of the material via B/,4. Future methodologies which use mixture laws for the nonlinearity parameter should take the corrected relation, Eq. (10), into account. •R. E. Apfel, "Prediction of tissue composition from ultrasonic measure- ments and mixture rules," J. Acoust. Soc. Am. 79, 148-152 (1986). 2C. M. Sehgal, G. M. Brown, R. C. Bahn, and J. F. Greenleaf, "Measure- ment and use of acoustic nonlinearity and sound speed to estimate compo- sition of excised livers," Ultrasound Med. Biol. 12( 11 ), 865-874 (1986). 3E. C. Everbach, "Tissue composition determination via measurements of the acoustic nonlinearity parameter," Ph.D. thesis, Yale University, New Haven, CT (December 1989). 4R. E. Apfel, "The effective nonlinearity parameter for immiscible liquid mixtures," J. Acoust. Soc. Am. 74, 1866-1868 (1983). 5p. L. Chambr6, "Speed of a Plane Wave in a Gross Mixture," J. Acoust. Soc. Am. 26, 329-331 (1954). 6One of the authors, Zhe-ming Zhu, has independently derived the result given in Eq. (10) by taking the pressure derivative of the effective com- pressibility [Eq. (8) ] and noting that the effective B/A is given by d( 1/ ,8)/dp - 1 (private communication). 7R. E. Apfel and W. N. Cobb, "Effective acoustic parameter (B/A) of a mixture," J. Acoust. Soc. Am. Suppl. 1 72, S40 (1982). 8M. F. Hamilton and D. T. Blackstock, "On the coefficient of nonlinearity /5' in nonlinear acoustics," J. Acoust. Soc. Am. 83, 74-77 (1988). 447 d. Acoust. Soc. Am., Vol. 89, No. 1, January 1991 Everbach ot al.: Letters to the Editor 447  Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP:  130.58.65.20 On: Thu, 21 May 2015 17:02:47

A Corrected Mixture Law For B/A

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A Corrected Mixture Law For B/A

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