# Ultrasonic velocity models in liquids (micro- and nanofluids): theoretical validations

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## Abstract

The theoretical ultrasonic velocity in any micro/nano fluids is computed in connection with the literatures. Kudriatsav theory (KT), Jouyban–Acree model (JAM), Floty theory (FT), Ramasamy–Anbanantham model (RAM), Glinkski model (GM), McAllister model (Mc-AM), time average model (TAM), Tavlorides groups (TG), Urick model (UM), Kuster and Toksoz model and modified Urick model (MUM) methods are computed. Further the validity of those theory to identify or check or confirm the possibility of existence of type of interaction, ideal and non-ideal behavior of the system was explained in the basis of hydrogen bonding interaction, induced dipoles interactions. The values have been computed for the pure solute and solvent with the binary, ternary, and pentenary mixtures above the marginal range of miscibility at various temperatures. The models/theory is relevant to the type of fluids and the medium such as ionic, electronic, micelle, aqueous or any type of liquids. The structure breaks the bonds in the associated molecules into their components by means of temperature. The measured parameters are fitted with their polynomial relations to compute the coefficients and standard errors for the validation of the experimental results. McAllister three bodies and multibody models were used to correlate their properties at various temperatures showing association/disassociation nature.

## Keywords

Ultrasonic velocity Theoretical models Liquid mixtures Molecular interactions Nanofluids## Introduction

The field of atomic and molecular acoustics relates the macroscopic parameters of sound waves and their properties of the propagating medium. And so, the studies of velocity and adsorption in sound give detail about the properties of gases, liquids such as in much different dispersion. The dependence temperature and concentration dependence of acoustic studies pave some significant observations of intermolecular interaction in liquid medium and solutions, since the liquid mixtures were found applications in the area of medical, technology, agriculture, fertilizers, restorations of food and related industries [1, 2]. In recent times, the ultrasonic analysis found wide applications in determining the physicomechanical (solids), physicochemical and bio-physical behavior of variety of liquid mixtures [3, 4]. The acoustical with thermodynamic properties evaluated in the in liquids mixtures, which are not obtained by any other methods which may be useful mainly in automobiles like combustion, petrol and diesel. Further, confirming the existence of specific interaction, the authors try to relate the values from experimental findings with those predicted molecular models and to know the molecular insight. This comparison gives fruitful information about the mixtures to test the validity of many empirical theories for the mixtures of binary, ternary [5, 6], tertiary and sometimes pentenary [4]. The thermodynamic dependence of fluids is more useful for sustaining the properties (density, viscosity) at high temperatures with more mechanical operations worldwide like train engine working continuously more than 100 h. The race car and high-speed motors run many revolutions per minute (tribology). It creates more temperature, absorbing heat from components of mixtures and remove the heat frequently by nanofluids. In recent time’s nano-lubricants, nano-coolant plays a vital role for high-speed mechanical operations.

The cohesive and adhesive properties of liquids indicate the nature of suspended solid particles and also indicate the intermediate properties of gas and solids. So, the liquid possesses the intermediate physical properties of solid and gasses, collectively called as condensed matter. The idea behind the liquid is considered as disordered solid and as condensed gas and so the pair potential parameter is accountable [7, 8].

Based on the molecular structure, Eyring proposed that the idea of condensed matter behavior and Hilderbrand commented and refused to consider a solid [9]. The molecular movement sometimes favors a condensed state, but the thermal energy agitates and destroys the actual structure. The concluding analysis is based on the molecular theories like perturbation, spherical particle, etc. [10]. And so, the acceptable partition function cannot be obtained due to many structures of particles other than sphere.

The nature of the molecules can be studied in liquids and liquid mixtures with statistical analysis from the standard references. Besides the available models, some new models can be proposed. However, the aim is to investigate the interaction, catalytic activity, molecular–kinetic nature for the assumed systems; it is much important to select and apply one model among those existing models. If the available models are not suitable to validate the experimental results, we can propose new model suitable for experiment with valid assumption. Evaluating ultrasonic velocity values computed from various models in liquid mixtures are compared and validate with suitable following model [11]. These models are Kudriatsav theory (KT), Jouyban–Acree model (JAM), Floty theory (FT), Ramasamy–Anbanantham model (RAM), Glinkski model (GM), Mc. Alister model (Mc-AM), time average model (TAM), Tavlorides groups (TG), Urick model (UM), Kuster and Toksoz model and modified Urick model (MUM). Also, the percentage deviation, standard percentage error and Chi-squared test are computed [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

While making the solution and for calculation of the percentage composition of the constituent of liquid mixtures [32, 33, 34], to find the strength of molecular interactions of liquid mixtures, through the non-rectilinear behavior of ultrasonic velocity. To obtain the additional information about the nature and strength of interaction of molecule, other parameters related to ultrasonic velocity can be used. The important features of ultrasonic velocity in experiment and the relevant experimental model are non-invariant, precision, rapidly and easy automation. Several researchers reported on ultrasonic velocity with models used for computing ultrasonic velocity [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

## Experimental

The Analar Grade chemicals are purified with normal procedures [32, 33]. And proper mixtures are prepared with mole fractions and the experimental ultrasonic velocities are measured with ultrasonic interferometer of any made, with an accuracy of Δ*U* ± 10^{−2} ms^{−1}. The density of liquid mixtures was determined with densitometer and the viscosity of the samples was measured with Ostwald’s viscometer with the accuracy of Δ*ρ* = ± 10^{−6} cm^{−3} and Δ*η* = ± 0.001 kg m^{−3}, respectively [34]. Further, it is completely free from air and bubbles (complete solutions). The particle sizes can be determined with laser diffractions/any other conventional methods suitable to the nature of solutions.

## Theories/models

### Kudriavtsev theory (*U* _{KT})

*U*

_{KT}, through the following relation:

*M*

_{A},

*M*

_{B}, and

*M*represent the mol. wts. of components

*A*and

*B*, the effective molecular weight

*M*

_{eff}= (

*X*

_{A}

*M*

_{A}+

*X*

_{B}

*M*

_{B}),

*L*represents the heat capacity of the mixture in cal/g.

### Jouyban–Acree model (*U* _{JA})

*j*:

Here, *U*_{1} and *U*_{2} are velocities of pure liquids 1 and 2. *X*_{1} and *X*_{2} are the mole fractions of 1 and 2 correspondingly.

### Flory’s theory (*U* _{FT})

A significant method from surface tension of mixtures is the Flory based on statistics [16]. Patterson and Rastogi proposed the model with reduced parameters [17]. The values of ultrasonic velocity from surface tension, temperature, pressure, volume with their reduced parameters through Auerbeck relation [35]

*K*,

*P** and

*T** are the Boltzmann constant, pressure and temperature of corresponding characteristics nature of fluids. Here

*α*’ is the thermal expansion coefficient and ‘

*k*

_{T}’ is the isothermal compressibility,

*V*through ‘

*α*’ is from the following expression:

*T** is given as:

*ψ*is the segment fraction,

*θ*

_{2}is the site fraction, and

*X*

_{12}is the interaction parameters:

*M*is the fraction of nearest neighbors, the molecule moving from the bulk to the surface. And the surface tension (

*σ*) of a mixture is

### Ramasamy and Anbanantham model (U_{RAM})

*K*

_{as}is

*A*] is the amount of solvent and [

*B*] is the amount of solute in the liquid mixture.

*X*

_{A},

*X*

_{AB},

*U*

_{A}and

*U*

_{AB}are the mole fraction of component

*A*, and associative

*AB*, ultrasonic velocity of component

*A*and associate

*AB*and observed ultrasonic velocity, respectively, and sometimes with non-equilibrium condition. With the existence of non-associated component, the above expression becomes:

*X*

_{B}and

*U*

_{B}are the mole fractions of component

*B*and ultrasonic velocity of

*B*(non-associated component).

*C*

_{A}and

*C*

_{B}are initial molar concentrations of the components. The value of

*K*

_{as}computes the equilibrium value of [

*AB*] for every composition of the mixture as well as [

*a*] = C

_{A}− [

*AB*] and [

*B*] =

*CB*− [

*AB*].

*a*

_{A},

*a*

_{B}and

*a*

_{AB}are the additivity of components

*A*,

*B*and associate

*AB*with equimolar which may be equal to

*a*′

_{A}and

*a*′

_{B}are the activities of [

*A*] and [

*B*] in equimolar quantities.

*K*

_{as}is

*AB*,

*U*

_{AB}, it is possible to equate the ultrasonic velocity calculated using Eq. (26) with the experiment.

*U*

_{obs}and

*U*

_{cal}are the observed and calculated equilibrium properties, respectively.

The low values of *S* can be obtained theoretically by a pair of fitted parameters. But, we found that for some *K*_{as} and *U*_{AB}, the values of *S* are high and varied rapidly, and for others, it is low and changes occur slowly when changing the fitted parameters. In these types, the value of *U*_{AB} is not much lower/higher than the lowest/highest observed ultrasonic velocity of the system. Quantitatively, it should be accepted with reasonable parameters *K*_{as} and *U*_{AB} which has the physical meaning and which reproduces satisfactorily experimental properties.

### Glinkski model (*U* _{GM})

*ϕ*of the components, the extracted form by Natta and Baccaredda model [20] as:

*U*

_{GL}is the theoretical ultrasonic velocity of binary liquid mixtures,

*ϕ*

_{A}and

*ϕ*

_{B}are the volume fractions of components

*A*and

*B*and

*U*

_{A},

*U*

_{B}and

*U*

_{AB}are the ultrasonic velocity of components

*A*,

*B*and

*AB*. The numerical procedure, computation and calculation of constant are same as described before. In addition to this, model which was already mentioned was further developed and elaborated by many researchers [21, 22, 37].

### McAllister 3-body model (*U* _{McA-3})

*G*

_{ii}, Δ

*G*

_{ij}or Δ

*S*

_{ijk}) of individual interactions, with their fraction and the total occurrences (\( x_{i}^{3} ,\;x_{i}^{2} x_{j} ,\;x_{j}^{2} ,\;x_{i} x_{j}^{2} \;\;or\;\;x_{i} x_{j} x_{k} ) \). Hence

### McAllister 4-body model (*U* _{MCA-4})

### Time average model (*U* _{tAvg})

### Tavlarides and co-workers model (*U* _{TM})

*g*

_{d}and

*g*

_{c}depend on the ratio

*γ*and subsequently the model produces branches depending on phase continuity [23, 24],

*γ*≥ 1

*γ*≤ 1

### Urick model (*U* _{Ur})

### Kuster and Toksoz model (*U* _{KUT})

### Modified Urick model (*U* _{MUr})

In this model, they account the scattering by thermal means, but applying the long wavelength limit with the two branches from phase continuity [27, 28, 29, 30].

*p*

_{m}and

*k*

_{m}, wee obtain

*α*

_{1}and

*δ*

_{1}as coefficients in front of

*ϕ*and

*ϕ*

^{2}terms);

## Validations

Among the all above models, the validations of some models were illustrated as follows, the time-average model, which equivalent to the sound propagation through a the medium is considered as layer by layer, (ii) In this modified time-average model by Taylaries and co-workers [23, 24], (iii) Urick model [25], (iv) model of Kuster and Toksoz [26] and (v) modified Urick model [27, 28].

Meng and his co-workers studied the above models for water with oil at temperatures 25 °C, 40 °C and 55 °C [29]. For the comparison of models (i)–(iv), model (i) based on the time-average approach is not a sufficient model to describe the behavior of oil–water mixtures studied here. Because in this model there is no consideration of any scattering effects, it is slightly concentrated with high dispersion media (droplets). The models (iii) and (iv) seem to match the experimental data most closely. The phase inversions of oil and water occur at the content values of 30–70% and that inversion causes due to the thermodynamic conditions. The application of model (v) is considering scattering by thermal energy of thermal with *β*, specific heat capacities, *C*_{p} and *C*_{v}, for both continuous and non-continuous medium.

The effect of specific heat capacities for hydrocarbons of petrochemicals like crude oil and water in this experiments reports is generally a mixture of many hydrocarbons and the properties are similar [30, 31, 38, 39]. From the model (v), the view of *θ* depends on the magnitude of factor (*μ *− 1), mostly which is small in case of fluids for water and very large molecules of hydrocarbons. The effect of temperature and the changes in *C*_{p}, *μ*_{water} = 1, *μ*_{oil} = 1 become *μ*_{water} > 1 and *μ*_{oil} > 0.

The Ramasamy and Anbanantham model [18] was corrected by Glinski [19] and tested by Natta and Baccaredda [40] to predict direct and the related associational behavior of liquid mixtures. The quantities analyzed were refractive index, molar volume, viscosity, free length, free volume, internal pressure, compressibility and many [35, 36, 37, 40, 41]. The results are fitted with the adjustable parameters. The values of *K*_{as}, *UA* and *B* are the fitted parameters. On changing the parameters, the equilibrium concentrations of individual components [*A*], [*B*] and [*AB*] change and the ultrasonic velocity can be computed. The difference in experiment and theory values for ultrasonic velocity is used to obtain the sum of squares of deviation. The values of ultrasonic velocity in pure associate can be treated as a fitted one with the values of *K*_{as}.

Values of (*α*) and *K*_{s} are needed in the PFP model obtained from the equations which have been tested already in many cases [36, 37, 41, 42, 43]. The advantage of the Flory theory over others is that the essential parameters required in this theory can be precisely determined using the physical parameters of pure liquids [44]; the average percentage deviations of the ultrasonic velocity obtained by the Flory statistical theory for both the mixtures have been discussed [45]. The ultrasonic velocities of both the mixtures were also analyzed in terms of the Flory statistical theory using Eq. (19). The APD of the comparison of ultrasonic velocity are reported in et al. [46] and it is found good agreement between the experimental and calculated values of u with a minimum APD of − 0.81 for the THF + 1-P mixture and a maximum APD of − 5.15 for the THF + 2-P mixture [46]. It may be concluded that the statistical mechanical theory can be used as a powerful tool to evaluate the excess functions directly from the ultrasonic velocity and density data.

*y*’ shows deviation in ultrasonic velocity,

*x*

_{i}is the mole fraction and

*A*

_{i}is the coefficient of

*i*th component, respectively. The coefficient values can be computed through multiple regression analysis (least square) and are summarized along with the standard deviations.

*Y*

^{E}denotes any excess values (excess ultrasonic velocity). The reduced Redlich–Kister (RR–K) functions give much better insight information of the non-ideality. i.e., RRK excess property is more sensitive than direct excess property occurring at low concentrations [48, 49]. The liquid mixtures having specific interactions like association at low concentrations with different size of molecules (clusters). And in the RRK- polynomials remove the effect of excess functions and gives specific reduced functions in (55) characterizing the velocity and other property and also gives evidence to the existence of important interactions [48, 49].

The solubility behavior of liquid mixtures with organic solvent is from *i* = 0 to *i* = 3 [50]. And it is also able to describe the multiple solubility maximum and solubility at various temperatures [51]. The miscibility and solubility are the key factors while designing the drugs in mixed solvents [52].

McAllister coefficient *a*, *b* and *c* were calculated using the least square procedure. With the variation in mole fractions, the values of speed of sound obtained from all the models decrease in trend at all temperatures.

The observations leading to the associated components parameters provide better agreement over the non-associated components’ parameters. It means that larger deviation values in PFP model can be explained as the model was developed for non-electrolyte γ-meric spherical chain molecules and the system under investigation has interacting and associating properties, Even though the formula used for this computation of *α* and *β*_{T} is also empirical in general. The molecular function and complex functions lead to positive deviations in speed of sound, whereas negative deviations are due to molecular dissociation. The sign and magnitude of deviations in sound depend on the relative strength of the two opposite effects. The less smoothness in deviations is due to the interaction between the existing component molecules. Isentropic compressibility shows increasing trend with the increase in mole fraction, and the density and ultrasonic velocity show other behavior [46].

The recently developed Juoyban–Acree model is applied successfully to the mixture of ortho, meta, para cresols and 1,4 dioxane, aniline and pyridine; these correlations in all six systems further fitted with Redlich–Kister Polynomials. From the analysis, Juoyban–Acree model is fitted better over than other studied models [53]. The spectroscopic analysis such as FTIR and UV–Visible absorption study can support to confirm the acoustical results obtained from various derived and theoretical parameters. And it is applicable to some alcohols and electrolytes [54, 55, 56, 57].

## Conclusions

Here, the author taken twelve no of theory/model of ultrasonic velocities for various liquid mixtures and are validated with experimental observations. The trends in observation of any one model suggest highly suitable with particular mixtures, either ideal, non-ideal and strength with magnitude of interactions of confirmed. Overall, further it is stated that all the models used in this present investigation successfully agree well with the experimental values and findings, and its shows that the liquids should have poor associating properties. In future, a general equation is fit to explain the liquid property more reliable and acceptable for associate and non-associated parameters of many fluids at nano and all ranges.

## Notes

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