Measurement of thermophysical properties of fluids

Measurement of Thermophysical Properties of Fluids I

R. Tufeu liThe experimental determination of the thermophysical properties of fluids reLIMHP--CNRS, Centre Universitaire Paris-Nord mains an active field of research. Experimental methods, whose principles are Villetaneuse, France in most cases classical, use modem electronics and computers. The introduction of these techniques frequently improves the accuracy of the measurements and in any case makes the experimentation less tedious. The principles of the methods are discussed and some devices are briefly described. Both equilibrium properties and transport properties are reviewed.

Keywords: measurement techniques, thermophysical properties, equilibrium properties, transport properties

INTRODUCTION Pure fluids and fluid mixtures are common substances in many chemical engineering processes. To design efficient equipment, data on fluids over a wide range of temperatures and pressures must be available. Due to the rapid advance of technology in new fields, there is an ever-increasing demand for accurate data. Though an enormous amount of data has been collected over the years, there remains a significant gap between demand and availability. It must be noted here, and users must be aware, that the data given in handbooks or references from data banks have not always been critically evaluated. Apart from widely investigated substances, the values of thermophysical properties are in many cases given with large (or withou0 error bars. This is due to the fact that most of the experimental values of these properties have been obtained by only one experimenter and consequently it is difficult to assign a figure to the accuracy of the measured values. In situations where no experimental data exist, the engineers and scientists dealing with fluids have at their disposal predictive methods to evaluate their thermophysical properties. These estimates are generally based on theories. Completely theoretical predictions are based on the molecular theory of matter and follow the "route royale," which goes from the intermolecular forces to the bulk properties by using the general concepts of statistical thermodynamics. Formal expressions of the thermophysical properties have been derived, statistical mechanics being used for equilibrium properties and kinetic theory for transport properties. Rigorous calculations of these macroscopic properties are possible in the case of an almost completely disordered assembly of molecules. The equilibrium properties are calculated for molecules interacting through intermolecular potentials of almost any complexity, while the transport properties can be exactly calculated only for molecules of spherical symmetry. For dense fluids, the solutions of the formal expressions are obtained through approximations (superposition approximation, Perkus Yevick, hypernetted chain approximations for equilibrium properties, and Enskog theory for transport coefficients) and analytically

calculated for some intermolecular potentials (rigid sphere, Lennard-Jones). The difficulties encountered in the calculation of thermophysical properties from general relations of statistical thermodynamics have been partly overcome by the development of computer simulation techniques. The art of simulating fluids has made enormous progress since the late 1950s and 1960s. Monte Carlo and molecular dynamics simulations have led to quantitative evaluation of equilibrium and transport properties of fluids even when sophisticated intermolecular potentials are used. The availability of sufficiently realistic molecular potential models for real fluids remains the main problem in generalizing the method of evaluating fluid properties. Another, no less important, problem is the high cost of computer time for such simulations. For these reasons, rigorous statistical theories appear to be of little help to engineers. However, they provide useful bases to propose semiempirical relations. The best-known example is the corresponding states approach, which allows the prediction of the unknown property of a substance in terms of the properties of a reference fluid. The semiempirical relations based on the principle of corresponding states associated with group contributions are widely used by engineers. An examination of these predictive methods, described in detail in Refs. 1 and 2, shows that they allow an estimation of equilibrium properties of pure fluids with an acceptable accuracy (a few percent) in the range of validity of the proposed correlations. The properties of mixtures are generally obtained with less accuracy. If the compositions of the coexisting phases of a mixture under pressure are accurately predicted (except close to the critical point), the evaluation of the densities of the phases is generally made with poor accuracy. The estimation of the transport properties is less precise than the prediction of the equilibrium properties (accuracy an order of magnitude lower is frequently observed). It must be noted that the semiempirical correlations allow the calculation of properties in terms of some macroscopic quantities that are not always experimentally available. In such cases, predictive methods have to be used to evaluate these

Address correspondence to Dr. R. Tufeu, LIMHP--CNRS, Centre Universitaire Paris-Nord, Avenue Jean-Baptiste Clement, 93430, Villetaneuse, France. Experimental Thermal and Fluid Science 1990; 3:108-123 © 1990 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010

108

0894-1777/90/$3.50

Thermophysical Properties of Fluids quantities. Consequently, errors are accumulated and it is difficult to give the accuracy of the predicted value. Moreover, the empirical correlations often contain adjustable parameters. To define these parameters, it is generally necessary to perform one or more experiments. From the above considerations, it is clear that experimental investigations of thermophysical properties of fluids with improved accuracy over extended temperature/pressure ranges cannot be avoided. It is difficult to give a complete description of all the methods and apparatus used to measure the thermophysical properties of pure fluids and fluid mixtures. The principles of the measurements of the considered properties and specificities of a few original devices recently proposed will be briefly recalled. E Q U I L I B R I U M P R O P E R T I E S OF F L U I D S

109

P CI

- - S - - C F

F

--LP

PVT Measurements F

Isodaoric Method The fluid of unknown density is contained in a cell for which the volume has been accurately determined as a function of temperature and pressure. The amount of fluid is obtained by different methods: •

•

It can be directly weighed [3] or weighed after transfer in an auxiliary container [4]. In the direct weighing method, it is now possible to weigh the cell very accurately with a pressure line attached to it by using electronic balances for which the vertical motion of the weighing pan is nearly zero. The fluid can be expanded into a large calibrated volume maintained at constant temperature (generally supercritical). The amount of gas in the expansion volume is obtained from the knowledge of the compressibility factor of the fluid at low pressure.

Variable-Volume Method The variation of the volume of the cell is obtained with a piston (dry piston [5, 6] or fluid piston [7]) or with a bellows [8]. See Fig. 1. A known amount of fluid is injected into the cell for an initial position of the piston or deformation of the bellows. The displacement of the piston, or of the end of the bellows, is noted as a function of the pressure externally applied to the cell. A calibration of the volume of the cell as a function of the displacement of the piston or of the deformation of the bellows must be done with fluids for which the equation of state is known with high accuracy. In the case of a dry piston, because of the unavoidable friction of the seals, the pressure must be measured with a gauge located in the cell (strain gauge, for example). The pressure variations can be automatically generated and the displacements of the piston or of the bellows recorded [9]. It must be noted that:

B

CF I Figure 1. Variable-volume P V T cells. (a) Dry piston cell. P, piston; S, seals. (b) Liquid piston cell. (c) Bellows ceil. F, fluid under experiment; CF, compressing fluid. The Burnett Method A Burnett apparatus (Fig. 2) consists of two cells of volume VA and VB maintained at constant temperature T. Cell A is filled with the gas at an initial pressure P1, and cell B is evacuated. Then the value between A and B is opened and the pressures are allowed to equalize between the two vessels. The expansion valve is closed, and B is again evacuated. The expansions are continued until a low pressure is reached. The result of the experiment is a series of decreasing pressures P1 . . . . . Pi . . . . . P f at constant temperature. The compressibility factors Zi satisfy the relation

(1)

Z i / Z 1 ~__p i g i - I /PI where N is the apparatus constant def'med as N = (V A + VB)/V A

(2)

With this method, the compressibility factor Z can be obtained with 0.1% accuracy [10]. The Burnett method provides the best measurements of densides less than half the critical density. V

EV

VV

1. The deformation of a bellows is not exactly reproducible. 2. When a fluid piston is used (for example mercury), the meniscus shape introduces errors on the volume of the cell, 3. The seals of the dry piston may absorb fluid under pressure. Moreover, their volume may change with time (extrusion). Variable-volume cells can be used to detect phase transitions in pure fluids as in fluid mixtures. The transitions are detected by analyzing the P(V)-r curve.

Figure 2. Burnett method. A, B, cells; T, thermostat; FV, EV, VV, valves.

110 R. Tufeu Buoyancy Force Devices This method is based on Archimedes' principle. In the devices that utilize magnetic suspension of a buoy in the fluid, a float that contains a magnetic is immersed in the fluid and acted upon by a solenoid. The inductance of a sensing coil depends upon the position of the magnet, and a servo circuit adjusts the current in the solenoid so that the Q of the sensing coils acquires a fixed value that corresponds to a fixed position of the float. The device must be calibrated with liquids of known density. Adsorption effects on the surface of the suspended buoy impose the ultimate limit on the accuracy of these devices. To improve the accuracy, the adsorption effects are compensated for by using two sinkers made of the same material; the two sinkers have equal weight and equal surface but a large volume difference. A two-sinker apparatus in which the buoyant force is directly measured by a microbalance has been designed by Wagner and co-workers [11, 12]. The density of pure gases and gas mixtures can be measured in the temperature range 0-50°C and in the pressure range 0.1-12 MPa with 0.015% accuracy.

Indirect Methods Refractive index and dielectric constant methods These methods are based on the Lorentz-

mospheric pressure, the filling of the tube must be done under vacuum when possible to avoid gas bubbles.

Sound Velocity Measurements The precise measurement of the velocity of sound in fluids as a function of temperature and pressure is useful in many respects. Sonic and ultrasonic measurements can be used to check the P V T and specific heat equations of state. Along with the PVTequation of state, the ratio of the specific heats as well as the specific heats themselves can be calculated as a function of temperature and pressure. From speed-of-sound data as a function of temperature and pressure and from the knowledge of the density p and the specific heat at constant pressure (Cp) along one isobar, and iterative method can be used to calculate p, Cp, the expansion coefficient ~p, the isothermal and adiabatic compressibilities, and the ratio 7 = C~/Cv. The starting relations are

an= ~ + ~ j

l

-e + 2 (P~)- ~ C te n2 n2

- -

OP

IT

= -T _

_t-~-T] r

(6)

(C.M.)

(3)

where u is the sound velocity and M is the molar mass of the fluid.

(L.L)

(4)

lnterferometric Methods The classical acoustical interferometer contains essentially a transducer that produces a

1

+ 2 (pc¢)-I _~Cte

(5)

and

Lorenz and Clausius-Mossotti laws relating the density p to the refractive index n and the dielectric constant e through the polarizability c~ of the fluid molecules: e-1

ctP

The devices (see the section devoted to dielectric constant and refraction index measurements) based on these methods make possible rapid measurements in the range where Burnett measurements become slow and tedious. They utilize small samples and short residence times in the apparatus, so they are advantageous for the investigation of toxic, reactive, or corrosive fluids. Extremely small variations of density can be detected with these methods (for example, the dielectric constant can be measured with a precision of 10-6), and consequently very precise relative densimeters can be designed. Vibrating tube densimeters Densimeters based upon the resonant frequency of vibrating U-shaped tubes filled with a fluid have been used and are available from commercial suppliers (for example, DMA densimeter, distributed by Anton Paar, Graz, Austria). These devices can provide densities precise to a few parts per million, but for highest accuracy the calibration of the apparatus must be done with great care. For this calibration, reference fluids of approximately the same density as the unknown fluid must be chosen. Vibrating tube densimeters are extremely well suited for the investigation of excess volume of mixtures. They also allow detection of phase transitions when, due to the formation of bubbles or drops, fluctuations in the resonant frequency are noted. The tubes are made of glass or metal. With metallic vibrating tubes, the useful range of the commercial devices has been extended to include pressure up to 35 MPa and temperatures up to 425 K. It must be noted that apparatus working under pressure must be calibrated not only as a function of temperature but also as a function of pressure. FOr liquid density measurements at at-

plane sound wave and a reflector parallel to the source. The incident and reflected waves interfere to form standing waves. Measurement of the amplitude variation in the sound field--say, location of two maxima or minima--allows the wavelength to be determined. The frequency being known, the velocity of sound in the fluid can be calculated. For fluids at low densities, sonic waves are used (/'equal to a few kilohertz). The generator is a diaphragm, and the detector is a microphone. For dense fluids, ultrasonic waves are used. The transducers and detectors are generally quartz crystals. Based on this technique, fixed path-variable frequency and variable path-fixed frequency apparatus have been developed [13-15]. An accuracy better than 0.1% in the determination of the sound velocity can be achieved. Measurements are generally made in tubes (circular cross section). It must be mentioned that the velocity of sound Utub¢ in a gas confined in a tube is less than that of the unconfined gas u. It is given by the Helmholtz-Kirchhoff relation U,ub¢ = U(1 -- ~ / V ~ )

(7)

where ~ is a function of the thermodynamic and transport properties of the fluid and of the diameter of the tube. The true sound velocity u is obtained by the intercept of the curve

u(1/veY).

The spherical resonator corresponds to the fixed-path interferometer class. In this case, a transducer generates sound waves in a spherical cavity. The speed of sound is determined by measuring the frequencies of the radial resonance. The transducer and the detectors (microphones) are located on the outer surface of the resonator; they are in contact with the fluid through small holes drilled in the wall of the sphere

Thermophysical Properties of Fluids

0

1

2

111

R

3

Bj

8

"VIV' "vlv" 'vlv' I

to

t~

t2

t3

Figure 3. Sound velocity measurements--pulse-echo technique. to, excitation; 1, 2, 3, echos. [16]. With this method, which demands high technical skill for the fabrication of the sphere, the velocity of sound at low and moderate densities can be obtained with high accuracy

(0.02%).

A,o

A0

[

Figure 4. Rayleigh (R) and Brillouin (13) lines of the light scattered by a fluid. frequency of the incident beam (Fig. 4). The latter is induced by the interaction between a photon and a phonon in the liquid under thermal equilibrium. The frequency shift Av between BriUouin peaks and the Rayleigh peak is given by

Av = =i=2z,in(u/c) sin 0 Pulse Methods The method most widely used for sound velocity measurements in dense fluids is the pulse technique. The cell is very similar to the single- or double-crystal interferometer. The continuous wave oscillator is replaced by a generator delivering short pulse voltages applied to a piezo crystal that vibrates at ultrasonic frequencies. The pulse travels over a distance d in the fluid and reaches a second quartz crystal (receptor) or the transducer again after reflection on a surface parallel to the transducer. Modern electronics allow very precise and accurate measurement of the transient time. An accuracy of 1 ns is easily obtained. So, for a velocity of 1000 m/s and a path length of 1 cm measured with 1 #m accuracy, the sound velocity is determined with 0.01% accuracy. For low absorption fluids in the frequency range of the transducer (some megahertz), the determination of the transient time can be more accurate because the path can be enlarged or the time corresponding to the nth echo can be measured. For measurements at high temperatures, the ultrasonic sound is transmitted to the fluid through buffer rods and the transducer and receiver are located at the ends of the rods maintained at room temperature [17]. In this configuration, measurements of the transient time with two fluid paths are necessary for good accuracy. In the echo overlap method (Papadakis method) [18-20], the repetition frequency of the generator is very low, so that all echoes corresponding to one excitation of the transducer die out before the next start (Fig. 3). The echo overlap is carried out optically on an oscilloscope by driving the time base at a frequency equal to the reciprocal of the time between two echoes. For precise measurements, the shape of the pulses (the envelope of the high-frequency oscillations of the transducer) must be well defined and not changed when reflected or when the temperature and density are changed. In the "sing-around system" [21], usually applied with a single transducer, the first returned echo of a train is used to trigger the pulse generator. The repetition rate is directly related to the transient time. Hypersonic Velocities of Sound in Fluids As predicted by Brillouin, the light scattered by a homogeneous liquid consists of a central Rayleigh line with the same frequency as that of the incident beam and two lines symmetrically shifted from the

(8)

where ~i is the frequency of the incident light, n is the refractive index of the liquid, u and c are the velocities of sound and light, and 0 is the scattering angle. The experimental system is composed of a laser, an optical cell, a Fabry-Perot interferometer, and a signal recorder. The method has been applied to measurements of sound velocity in molten salts [22]. By comparison of sonic and ultrasonic sound velocities, the dispersion of sound can be studied.

Heat Capacity Determinations and Measurements of Enthalpy Changes at Phase Transitions In calorimetry we are dealing with the heat involved in a known change in the state of a substance. This can involve changes in phase, temperature, pressure, and volume. Many kinds of calorimeters have been constructed. It is quite impossible to give a complete description of the devices that have been presented in the literature or are commercially available. All the efforts have consisted of attempts to approach ideal calorimeters or to improve the methods for accurate control and estimation of the corrections.

Isothermal Calorimeters The isothermal calorimeters are those in which there is ideally no temperature change during the experiment. Many isothermal calorimeters utilize a phase change to maintain the temperature constant. Using the drop method, the calorimeter measures the enthalpy change between an initial state and the final state 0 = Of from the amount of solid transformed into liquid. The prototype of the isothermal calorimeter is the Bunsen ice calorimeter. A serious disadvantage of the ice calorimeter is its rather low sensitivity (volume change of the ice-water mixture per unit of heat), so calorimeters using pure organic compounds have been used. The specific heat is obtained by differentiation of the enthalpy curve. This method is suitable and accurate for specific heat measurements of liquids at high temperature and atmospheric pressure (molten salts, fused metals) [23]. In the drop method the heat losses by radiation are small because they occur only during the short travel time from the oven to the calorimeter.

112 R. Tufeu

i

Sh

/f

:/_.z_--7 / :....':..../';" e H

In~ ~

HFD

Figure 5. Calvet heat flux calorimeter. V, vessel; S, sample; CB, calorimetric block; TC, temperature controller; In, insulation; HFD, heat flux detector.

Heat Flux Calorimeters In heat flux calorimeters, the heat generated in the sample is transferred to a calorimetric block that functions as a heat sink. The amount of heat is measured with a heat flux detector. Calvet calorimeters (Fig. 5) are based on this principle. They can operate up to 1500°C. In the most elaborate calorimeters, a heat flux as low as a few microwatts can be detected (microcalorimeters). Compensation methods are used for absolute measurements (two-cell calorimeters). Working at constant temperature, these calorimeters are adapted to measure the heat of reaction or the heat of mixing. (distributed by SETARAM in France).* If a temperature variation is imposed on the calorimeter, the flux meter indicates the amount of heat required to heat up the sample, and the specific heat of the sample can be measured. The specific heat of liquids along the saturation curve can be measured with this apparatus. The microcalorimeter makes it possible to work within an extremely small temperature interval; consequently, the correction for evaporation of the liquid sample becomes negligible and the heat capacity measured is the true value at the temperature of the calorimeter. As an example, the specifications of a commercially available heat flux calorimeter working at a constant heating rate are the following: Temperature: ambient to 300°C Volume of the cell: 15 cm3 Sensibility: 15 #W Accuracy: 0.5% Adiabatic Calorimeters Calorimeters in which there is no heat transfer across their boundaries are called adiabatic calorimeters. Practically no calorimeter is truly adiabatic. The ways to approach adiabaticity are the following: •

• •

Reduce to a minimum the temperature difference between the calorimeter itself and the surrounding shields. Minimize the heat leakage by conduction and radiation. Minimize measurement time. The time needed to reach the temperature equilibrium of the calorimeter must be as short as possible (compared to the time constant of equilibrium of calorimeter and surroundings).

* SETARAM, 7 Rue de l'Oratoire 69300 Calluire, France.

Figure 6. Flow calorimeter. T,., Tf, initial and final temperatures; IV, input power; Sh, shields. For the fabrication of adiabatic calorimeters, the main efforts go into reducing the heat leakage either by controlling the temperature of the surrounding surface of the shields, which are set at the temperature of the calorimeter, or by using shields with a low emissivity coefficient (highly polished silver or gold plates, for example) and avoiding a low thermal resistance between the shields and the calorimeter (employing a vacuum and suspending the calorimeter by long thin wires). These calorimeters are well adapted to measure Cv and the specific heat Cs of the saturated fluid. However, at high pressures and generally for fluids having a low compressibility, for Cv measurements a large part of the heat needed to increment the temperature of the calorimeter + fluid is used to increase the temperature of the calorimeter (including the liquid container) itself [24]. Due to the increase in the heat transfer by radiation (T3 A T function), the difficulties encountered in making a perfectly adiabatic calorimeter increase with temperature. A flow calorimeter is a calorimeter that is used with fluid samples flowing through it (Fig. 6). A known amount of heat per unit time is injected in a constant fluid flow entering the calorimeter at the initial temperature Ti. The specific heat (more precisely, the enthalpy change) is calculated from the input power W, the temperature increase of the fluid T / - Ti, and the fluid flow m. In this kind of calorimeter the heat losses are difficult to evaluate and to control. Calibrations are often necessary. Assuming that the heat losses are a function of the temperature increment in the gas only, the true specific heat of the fluid at constant pressure Cp is obtained by extrapolation of the quantity W/m measured at different fluid flows m to 1/m = 0, the beating power W being adjusted to maintain the temperature increase constant. However, generally, the heat losses are also dependent on flow rate. Another difficulty with these calorimeters is the measurement of the temperature of the fluid for high fluid velocities. Flow calorimeters are used mainly for accurate measurements (within a few percent) of the specific heat of gases at constant pressure. One of the best examples of the use of a flow calorimeter is the measurement of the specific heat Cp of steam at high temperature and high pressure [25]. An extensive presentation of calorimetric methods can be found in Ref. 26. E Q U I L I B R I U M PROPERTIES OF M I X T U R E S

Phase Equilibria in Mixtures Many separation processes utilize phase equilibria. Consequently, among the thermodynamic properties of fluid mixtures, the phase equilibrium properties are especially importam. Experimental methods for the determination of phase equilibria produce data that are immediately valuable to chemical engineers.

Thermophysical Properties of Fluids Measurements of phase equilibria in mixtures is a vast subject by itself and we will only touch lightly on the topic. In mixture thermodynamics, it is often practical to distinguish between so-called density variables (mole fractions, molar volumes) and field variables (pressure, temperature). The experimental problem is, of course, the determination of the density variables. Determination of Mole Fraction Synthetic methods A mixture of known composition is prepared and its behavior is observed as a function of temperature and pressure. The volume of the cell is varied with a piston (dry or liquid piston). Generally, windows allow observation of the phases. Cells made of a sapphire tube operating up to 50 MPa and allowing observation of the entire sample are now available. In a typical experiment, the temperature and pressure are adjusted so that the mixture in the cell is in a one-phase region. Then the temperature (at constant volume) or the pressure (at constant temperature) is decreased until a new phase forms. The pressure, the temperature at which the phase separation begins, and the mole fraction of the prepared mixture define a point of the phase envelope [27, 28]. There are some experimental difficulties in detecting the phase transition (mainly the dew point). This determination can be achieved by various methods. ,

•

•

•

The P O O r and P(T)v curves generally show a more or less well marked change in their slope at the transition [29]. However, when the concentration of one of the components of the binary mixture is small 0ess than a few percent) or when the experiments are performed in the vicinity of the critical point of the mixture, this change can be so small that it cannot be observed and other methods of detection must be used. The transition can be detected with a laser beam passing through the fluid sample. At the transition a "fog" is formed in the bulk of the fluid and the scattered light is seen in the cell with a photodetector. Another technique uses a microwave apparatus. It measures small shifts in a static microwave energy field caused by the appearance of a second phase [30]. A capacitance can also be used. Very small variations in the dielectric constant related to the change in composition may be detected (see refractive index and dielectric constant measurements).

The synthetic method allows the determination of characteristic lines (critical lines, liquid-liquid-vapor) of the phase diagram. For the determination of the critical parameters Pc, Vc, Tc of a mixture at a fixed composition xi, the volume is varied until the transition from one phase to two phases occurs so that the appearance/disappearance of the meniscus is observed in the middle of the cell with strong light-scattering effects. Analytic methods These methods are suitable for multicomponent mixtures. FOr binary mixtures, they do not have to rely on precise knowledge of the overall composition. Here, temperature and pressure are adjusted to bring out a phase separation. Samples are taken from the phases and analyzed [31]. The main difficulty in this method lies in the preparation and handling of the samples. The mixtures are prepared in a constant-volume or variablevolume vessel. The advantage of a variable-volume cell is that the pressure can be kept constant during sampling, and conse-

113

quently the equilibrium is not perturbed, a condition that must be fulfilled near the critical point. When a circulation method is used, one or both of the components are continuously injected into a cell where the coexisting phases are formed and continually withdrawn and analyzed [32]. Many sampling techniques have been used. Examples are expansion into the sample loop of a six-way valve and transfer to the analytical device and the use of detachable sampling vessels. The compositions of the samples are generally determined by gas chromatography. In the circulation method, when one of the components has a low volatility (liquid at normal conditions) while the other is a gas at normal conditions, the compositions of the phases can be obtained by expansion, measurement of the volume of gas, and weighing of the condensed heavy component. To avoid sampling, in situ determination of the composition of the coexisting phases would be preferable. In principle, spectroscopic methods can be used; these include ultraviolet [33] and infrared absorption [34] and Raman spectroscopy [35]. The difficulty with these methods is that extensive calibrations are needed and they can be used with only a restricted number of compounds. For example, the application of ultraviolet light spectroscopy is more or less restricted to aromatic or colored compounds. Densities of the Phases For a severe and thorough test of theoretical calculations of phase equilibria, both phase compositions and densities are needed. When two or more phases are present, their densities can be obtained by extracting sampies of known volume in the different phases. In the case of binary mixtures, the knowledge of the volumes occupied by the two coexisting phases, for two fillings of the equilibrium cell at different overall compositions and at constant temperature and pressure, allows the determination of the densities and the compositions of the two coexisting phases [36]. A more complete review of recent papers on phase equilibria can be found in Ref. 37. Excess Properties of Mixtures Excess Molar Volume The excess molar volume Ve of a binary homogeneous mixture is defined by Ve(P, T, x) = Vm(P, T, x) - x V l ( P , T) - (1 - x)V2(P, T) (9) Here x and V 1 a r e the molar fraction and molar volume, respectively, of component 1 and V2 is the molar volume of component 2. Consequently, Ve can be obtained from P V T measurements of the homogeneous mixture and of the pure fluids. All of the methods described to measure the equation of state of pure fluids can be applied to mixtures. When a variable-volume method or the Burnett method is used, one must take care that no phase separation will occur. Direct measurement of Ve can be performed with liquids at atmospheric pressure by using simple picnometers. Excess Molar Enlhalpy [38, 39] The definition of excess molar enthalpy H e is identical to the definition of molar excess volume. H e can be obtained directly by measuring the heat of mixing of the components. The heat flux calorimeter is well adapted

114 R. Tufeu 14

HP

r,

Figure 7. Thermal conductivity--parallel plate method. CP, cold plate; HP, hot plate; GR, guard ring; H, heater. for the determination of the heat of mixing of liquids (at atmospheric pressure). Due to experimental difficulties, results for high pressures are very rare. Recently, accurate high-pressure flow calorimeters have been developed that are suitable for measurements of endothermic and exothermic H ~. TRANSPORT PROPERTIES OF F L U I D S The transport properties of fluids have been recognized as very important for the chemical industry, because heat, mass, or momentum transfer are very often involved. Their values are widely used to design heat exchangers, compressors, flowmeters, pumps, distillation columns, and other equipment. Thermal Conductivity Measurements The thermal conductivity coefficient is phenomenologically defined as the coefficient of proportionality that relates the conductive heat flux to the temperature gradient (Fourier's law): Q = -)x grad T

(10)

The methods used for the determination of thermal conductivity are the same for liquids and gases.

Steady-State Methods In steady-state methods, the partial derivative of the temperature T versus time is zero. The parallel-plate method employs a thin horizontal layer of fluid enclosed between two perfectly plane surfaces mainmined at two different temperatures (Fig. 7). Guard rings are used to ensure unidirectional heat flux in the vertically downward direction to avoid convection [40]. The plates must be perfectly parallel and the temperature difference between the guard ring and the hot plate must be controlled to be as close to zero as possible. This apparatus is delicate to make. It can work in large temperature and pressure ranges. Measurements from cryogenic temperatures up to a few hundred degrees Celsius and up to 250 MPa have been reported with an accuracy of the order of 1-2%. The method allows a relatively close approach of the critical point (a few tenths of a degree) [41]. In the concentric cylinder method, the fluid under experiment is located in the annular gap between two coaxial cylinders (Fig. 8). To avoid convection, the thickness of the gap is of the order of a few tenths of a millimeter. If the Rayleigh number (Pr × Gr) is less than 1000, the heat transfer by convection can be considered to be negligible. The heat is emitted in the inner cylinder, and a radial temperature gradient is established. Absolute values of the thermal conductivity of the fluid are obtained by using guard cylinders, which eliminate end effects. Thermal conductivity cells with-

OC IC

Figure g, Thermal conductivity--concentric cylinder method. IC, inner cylinder; OC, outer cylinder; GC, guard cylinders;/-/1, /-/2,/43, heaters. out guard cylinder cells have also been used [42]; part of the

power emitted in the inner cylinder is transmitted to end pieces at the same temperature as the outer cylinder. The cell must be calibrated to correct the apparent thermal conductivity for heat losses by the pins centering the inner cylinder and the electrical and thermocouple wires. The concentric cylinder method allows thermal conductivity measurements at temperatures higher than with the parallelplate apparatus because it is easier to maintain the initial geometry. As for the parallel-plate apparatus, the mechanical resistance of the high-pressure vessel containing the cell is the only limitation of the pressures at which the measurements can be performed. Measurement up to 1 GPa have been published. A 2°7o accuracy can be achieved with this technique. When a small-diameter wire is used instead of the internal cylinder in the coaxial cylinder method, the latter is called the hot-wire method. Here the wire is used as the heater and the internal resistance thermometer [43]. The temperature difference between the wire and the outer cylinder can be very high (up to 3000 K). This technique (the hot-wire thermal diffusion column) has been used to measure the thermal conductivity of gases at low pressure (P < 1 atm) [44]. When measurements are performed on electrically conducting fluids, the wire must be coated [45]. The geometry of the concentric sphere method is theoreticaily preferable to parallel plates because it avoids the inevitable thermal losses around the periphery of the plane surfaces of the parallel plate apparatus or at the ends of the inner cylinder in the coaxial cylinder method. However, the mechanical realization of cells based on this method is very difficult, and one cannot avoid heat losses through pins and wires. In fact, an accuracy better than 3% cannot be obtained. With all the above methods, to obtain the true values of the thermal conductivity a number of corrections must be applied. The correction for heat transfer by convection has been mentioned. Correction for heat transfer by radiation must also be taken into account. These corrections are not simple to evaluate when the fluid is not transparent to the radiation; here the equations of heat transfer by conduction and radiation are coupled. To make the corrections as small as possible, the cell must be realized with a material having a low emissivity coefficient (highly polished silver, for example).

Thermophysical Properties of Fluids In the case of the hot-wire method, the wire is usually made of platinum or tungsten, so it is necessary to choose a wire with a diameter as small as possible. At low densities, when the mean temperature of the molecular layer at the wall is different from the temperature of the wall, accommodation effects play an important role. Generally the thermal conductivity of very dilute gases (P <_ 1 atm) are obtained by extrapolation of values of the thermal conductivity measured at higher pressure. Precautions that have to be observed in the design of thermal conductivity apparatuses (which remain research laboratory apparatuses) and corrections that have to be applied have been discussed extensively in the literature [46--48]. Transient Methods In these methods, the principal measurement is the temporal behavior of the temperature field in the fluid. We will limit our. presentation to the transient hotwire method. Actually, this method has become accepted as the most accurate technique for the measurement of fluid thermal conductivity. The principle of the method is simple: The temperature rise A T as a function of time t of an infinitely long thin wire of radius a immersed in a fluid of infinite extent generating heat at a rate q is given by AT=~

q

4kt a2C

In--

(11)

where h is the thermal conductivity of the fluid, k is the thermal diffusivity, and C is a numerical constant. Equation (11) becomes operative after a transient period that follows the initiation of heat generation in the wire. )~ is determined by means of linear regression in a set of (AT, In t) data (Fig. 9). In practice, the temperature rise of the wire is determined from its resistance change, which itself is measured by means of an automatic direct current bridge [49, 49a]. It is usual to employ two wires that differ in length to take into account end effects. For high accuracy, the setting of the wire (,-~ 5 #m in diameter) must be made with great care to ensure that the wire is maintained under tension of a known amount over the entire defined temperature range of the measurements. Long studies of the shape of the spring used to stretch the wire without inducing vibrations have been performed [50]. The associated electronics allow precise measurements of the resistance of the wire as a function of time and control the generation of a constant heat q in the wire. A 0.3% accuracy can be expected with the most sophisticated apparatus in relatively large temperature and pressure ranges (from cryogenic temperatures up to 150°C). This technique does not appear to be well adapted for measurements in the critical region because the necessary increment in temperature for a measurement is too large. By coating the wire (with tantalum oxide or polyester, for example) at the cost of a loss in accuracy, this method has been used to measure the thermal conductivity of electrically conducting substances (electrolytes, molten salts) [51]. It must be noted that in contrast to the steady-state methods, where the convection cannot be completely avoided but is limited by using a thin fluid layer and small temperature gradients, the analysis of the behavior of the curve of A T as a function of In t allows detection of the onset of convection (see. Fig. 9).

115

T

0

e

0

0

0

~

0 9 O e 0

D6

0

B

Int Figure 9. Temperature increase of the hot wire as a function of the natural logarithm of time. The deviation from a straight line indicates the onset of convection. Thermal Conductivity Determination from Thermal Diffusivity Measurements The decay time r of the density fluctuations in a pure fluid is related to the thermal diffusivity Dr = k/pC p

(12)

1/r = Drq 2

(13)

by

where q is the wave number of the scattered light. This method, which is well adapted for experiments in the vicinity of the critical point of pure fluids has been extended to measurements outside this region. It uses a laser beam as the light source and a correlator (digital or analog) to measure the characteristic time of the autocorrelation function of the scattered intensity (in fact, the autocorrelation function of the photocurrent or the photopulses delivered by a photomultiplier tube used as a detector) (Fig. 10) [52]. The accuracy is generally poor ( ~ 5%). Another light-scattering technique (forced Rayleigh scattering method) has been proposed (see mass diffusion measurements). Viscosity of Fluids The viscosity of fluids extends from 10 -5 Pa.s for dilute gases to 104 Pa. s for very viscous liquids (molten salts). There are only a few apparatuses suitable for accurate measurement of the viscosity of fluids over such a wide range. The order of magnitude of the viscosity of the fluid will determine the most appropriate method to be chosen. The Capillary Flow Method This method is based on the Poiseuille law, which describes the theory of fluid flow

LB ~ S a

Figure 10. Scattered light experiment. Sa, sample; LB, laser beam; SL, scattered light; PM, photomultiplier tube; C, correlator; O, scattering angle.

116 R. Tufeu C •, .

:....

• ~,.

.,., ; 1.:," ; ::

:t

Figure 11. Viscosity measurement--capillary flow method. 12, capillary tubing. through capillary tubing (Fig. 11). The mass rate of flow mi is given by mi -

ra4 p(pi - Po) 81rl

(14)

Here a is the radius of the tubing, 1 is its length, p is the average fluid density, and P i and Po are the pressure at the inlet and outlet, respectively. Because of possible slippage of the fluid at the walls and the deformation of the flow profile at the inlet and outlet of the capillary, corrections must be applied to the simple law (14) to obtain ~1. For better accuracy, the radius of the capillary tubing must be carefully chosen. The constant-volume viscometer is a more precise capillary flow viscometer that allows the determination of the absolute viscosity of fluids with an accuracy better than 1% over wide ranges of temperature and pressure [53-55].

Oscillating Disk Viscometer This is a particular type of oscillating body viscometer. It consists of a suspended, axially symmetric body that performs torsional oscillations in the fluid (Fig. 12a). The principle of this method, which has been used mainly for gases and low-viscosity liquids, involves the measurement of the period and amplitude of the damped oscillations of a solid disk suspended at the end of a wire in the fluid. An exact theoretical analysis of the velocity field around the oscillating disk is not simple. Discontinuities are created at the vicinity of the edges when vertical and horizontal faces

°\

meet, increasing the viscous torque by an edge effect; however, this apparatus permits absolute viscosity measurements to be made. For an excellent accuracy of the instrument, the fixed plates and disk must be perfectly flat and parallel. A typical instrument capable of measurements up to 60 MPa consists of an optically finished quartz disk, with a radius of 7 cm and a thickness of 0.84 mm, suspended by a platinum-tungsten wire 5 × 10 -2 mm in diameter and 0.24 m in length [56]. Measurements of the velocity of gases with a 0.1% accuracy in the temperature range of room temperature to 473 K have been reported. At higher temperatures (measurements up to 973 K in dilute gases have been reported) the accuracy is not as good but remains better than 1% [57]. Oscillating Cup Viscometer The principle of the method consists in the measurement of the damping time of a cup containing the fluid suspended at the end of a torsional wire (Fig. 12b). This method has been used mainly to measure the viscosity of liquids at high temperature (molten salts and metals) [58]. Its accuracy is difficult to estimate (probably 2%). It has been shown that, due to the meniscus height, the measured viscosity depends on the level of the fluid in the cup. Falling Body Viscometers These viscometers are used mainly for liquids and dense gases. The principle of the method is based on Stoke's law. The apparent weight of a sphere falling in an infinite homogeneous fluid at a uniform velocity v is related to the viscosity ~ by W = 6~r~lav

(15)

where a is the radius of the sphere. This relation is valid only for extremely low Reynolds numbers, and corrections have to be applied to take account of deviations from this condition. Moreover, wall corrections may be needed. For falling bodies having different shapes (cylinder, cylinder with hemispherical ends, needle), the viscosity is obtained from the measurement of the time t for the weight to fall through a fixed distance at constant velocity. (16) The constant C depends on the dimensions of the apparatus and is determined by measuring the viscosity of a fluid of known viscosity. The viscosity and density of the fluid used to calibrate the apparatus must be as close as possible to the viscosity and density of the unknown fluid [59]. It is worthy mentioning that a complete Stokes type of solution for the falling needle viscometer (long cylinder with hemispherical ends) has been presented [60]. Consequently, this instrument allows absolute determination of the viscosity of fluids. rl = C (Ps - p f )t

"lI/lllll//llI////¢

Tw~OC

b

IC

¢

Figure 12. Viscosity measurements. (a) Oscillating disk viscometer. D, disk; TW, torsional wire. (b) Oscillating cup viscometer. C, cup; TW, torsional wire; L, liquid. (c) Rotating cylinder viscometer.

The Rotating Cylinder Method (Couette) Rotating instruments are by far the most popular in engineering. In this method the fluid is enclosed in the gap between two coaxial cylinders (Fig. 12c) [61]. When the outer cylinder rotates at constant angular velocity ~, the inner cylinder suspended by a torsional wire undergoes an angular deflection 4. If ! is the length of the inner cylinder, 1 its moment of inertia, ri its radius, and re is the radius of the outer cylinder, the viscosity is given by ~? =

x c b I (r2e - r~) r~r2eZZtol

(17)

Thermophysical Properties of Fluids

117

si

E.O.

O.O"

II

Figure 14. Block diagram of the wire viscometer. 1, DC supply; 2, mercury relay; 3, wire; 4, magnet; 5, low noise amplifier and fdter; 6, oscilloscope.

Figure 13. Quartz crystal viscometer. QC, quartz cylinder; El, electrode; H, holder; E.a., electrical axis; O.a., Optical axis. where z is the period of the free oscillations of the inner cylinder. In order to obtain absolute measurements, guard cylinders must be set above and below the suspended cylinder. The stability of the flow between the cylinders depends on the value of the appropriate Reynolds number Re = 6ore(re r i ) / v , where ~ is the kinematic viscosity. A sensitive method of testing rotating-cylinder viscometer measurements is to plot the effective viscosity over a range of rotation rates. An accuracy better than 1% can be obtained over large temperature and pressure ranges by choosing correct values of the gap re - ri. In its simplified form, due to the difficulty of correcting for end effects, this type of apparatus is used as a relative measurement device.

to the imperfection of the crystal and deviation from ideality (perfect alignment between the optical axis of the crystal and the cylinder axis; floating in the fluid with no ends); another part of the 2% is due to the uncertainty in the density of the fluid since the apparatus measures the product *lP [62, 63]. The torsional crystal oscillator was first applied to viscosity measurements of liquids. With refinements in the suspension of the crystal, this apparatus was adapted for measurements in low-density fluids and at high pressure. All of the instruments previously described are limited to a given range of the phase diagram and can seldom be used outside of it. An advantage of the torsionally vibrating viscorneter is that it allows the measurement of viscosities ranging from 20 × 10 -6 Pa. s to a few pascal-seconds.

Vibrating Wire Viscometer The vibrating viscometer method consists essentially of determining the attenuation factor of decreasing oscillation amplitudes of a tightened wire in a fluid (Fig. 14). A magnetic field allows the vibrations to start and to be converted into an electrical signal. The relation used to determine the viscosity ,1 is deduced from Stokes's phenomenological law. It is written fz

Torsionally Vibrating Quartz Viscometer The torsional viscometer consists of a cylindrical piece of piezoelectric material that can be driven in a torsional mode by the application of an alternating electric field (Fig. 13). In the medium surrounding the crystal this twisting motion generates shear waves that are very rapidly attenuated. From simple hydrodynamics theory, expressions relating the product of the viscosity ~ and density p to measurable resonant parameters of the crystal can be derived. Assuming that the crystal is driven in a purely torsional mode of vibration, the following expression is obtained:

"fff(M)2(ff

r/= ~-

Afvac~ 2 fvac ]

(18)

where A f is the bandwidth of the resonant curve at halfheight; M , S, and f are the mass, surface area, resonant frequency, respectively,of the crystal;and Af~c and f,c are the corresponding values in vacuum. The viscosityis determined with high precision (some tenths of a percent) if the crystal frequency (--.50 kHz) is stable and readable to one part in 10s with a resolution of 10 -3 Hz. By comparison with resultsobtained with other absolute methods, the estimated accuracy is 2%. Part of this2 % is due

= (llrk')(ps/pf

+ k)

(19)

where z -- 1/et, a is the attenuation factor, f is the frequency of the wire, Ps and ,of are the densities of the wire and fluid, respectively, and k and k ~ are two functions depending on a parameter m, m ~- (a27rfp//2~7)I/2

(20)

where a is the wire radius. The vibrating wire viscometer, because of the simplicity and compactness of its measurement cell, is well adapted for extreme physical conditions. It can be used for a wide range of densities from dilute gases up to dense fluids. Typical wire diameter and length are 200 #m and 50 mm, and the resonance frequency is about 2500 Hz. The estimated accuracy of the vibrating viscometer is 1.5-2% for fluid viscosities ranging from 200 to 5 x 105 #P (20 #Pa. s to 50 mPa. s) [64, 65].

Diffusion Coefficient Measurements Compared to the other transport coefficients, the diffusion coefficient has been less investigated. However, many methods have been proposed for its experimental investigation. Only a few of them will be briefly described here. Several effects associated with diffusion must be taken into

118

R. Tufeu t=O

I

x=O

I

Figure

15. Diffusion

il It. . . . . coefficient

r-"

! 1___ measurement--Loschmidt

method. account in diffusion coefficient measurements: the Dufour effect and volume and enthalpy changes in mixing. To minimize the corrections, it is necessary to maintain the nearequilibrium state by using small concentration gradients, and the experiment must be carried out under severe isothermal conditions to minimize the bulk flow motion. The classical methods that can be used to measure the mutual diffusion in a binary mixture can in principle be adapted to determine the self-diffusion in a pure fluid. In this case it is necessary to "put a mark" on some molecules; this is done with isotopic radioactive tracers. Loschmidt Technique The diffusion cell consists basically of two symmetrical sections A and B joined together about a common pivot (Fig. 15). Each half is composed of a cylinder closed at one end and attached to a plate in an off-center position at the other. The two isolated sections are fdled with mixtures of different initial compositions xi(O) and x[(0). At time t = 0, the two cell compartments are aligned (AB t) and are isolated again at time t. The new average compositions of the mixtures in each compartment are determined. The mutual diffusion coefficient is given by

1

O = ~ In

N

,,o 1

A

I

B

t= t

fluid

(Ca --CA)t

(21)

(Ca - CA)0

where/3 is a geometrical constant. The mixture compositions Cs and CA can be measured by sampling or in situ analysis. In another technique, the concentrations are measured as a function of time in two planes normal to the vertical axis and equidistant from the cell central plane. If the distance of the two planes from the center of the cell is chosen to be 1/3, the concentration difference is given by AC 2(t) : A exp

( Tr2D12t~ ff

j

(22)

The compositions may be obtained by using an interferometric arrangement. The main problem with this method arises from the way whereby the initial conditions are created. The connection of the two parts of the cell always causes a perturbation in the diffusion process [66].

The Diaphragm Cell This technique differs from the preceding one by the introduction of a porous diaphragm between the two compartments of the cell [67, 68]. The Taylor Dispersion Technique The ideal instrument for the measurement of the diffusion coefficient by the technique

Figure 16. The ideal Taylor diffusion experiment. of Taylor dispersion consists of an infinitely long straight tube of uniform circular cross section through which a fluid mixture with composition-independent physical properties passes in laminar flow (Fig. 16). Following the introduction of a function pulse of a mixture of different composition at a particular axial location in the tube, the combined action of the parabolic velocity profile and molecular diffusion induces diffusion of the injected fluid. The diffusion coefficient is determined from the distribution of the concentration perturbation at a distance L downstream from the injection point. The governing differential equation for the process is given in Ref. 69. A complete analysis of the departure of a practical instrument from ideality has been developed. It takes into account the sample introduction, the real diffusion tube geometry, and the concentration-dependent fluid properties. Various methods can be used to determine the concentration profile at the end of the tube (differential refractometer, for example). This method seems to be well adapted for liquids at atmospheric pressure [70]. The Spin Echo Technique The NMR spin echo measurement of self-diffusion is a sophisticated technique that can be used for fluids constituted of atoms whose nuclear magnetic resonance is within the frequency and field limits of NMR apparatus. The spin echo technique was originally used to measure the natural spin transverse relaxation time. The reader will find a clear description of the method in Ref. 71. It appears that two methods (simple and multiple exposure) lead to different values of the apparent decay time. The observed discrepancies have been explained by the effect of molecular diffusion. This effect is seen because of inhomogeneities in the applied external field. By controlling the external field it was possible to measure simultaneously the transverse relaxation time T2 and the diffusion coefficient D [72]. The maximum amplitude of the spin echo at time t = 2g" is given by A(2~-) = ct exp

T2

"y2GED~'3

(23)

where 3' is the gyromagnetic ratio (ratio of the magnetic moment to the nuclear angular momentum), G is the external field gradient, ~" is the time between the two successive radiofrequency pulses inducing a 90 ° rotation of the net magnetic moment initially parallel to the direction of the strong external field. For precise measurements of D, it is necessary to apply an external field gradient much larger than the inhomogeneities of the static field.

Thermophysical Properties of Fluids 119 Forced Rayleigh Scattering Method In this method [73], one measures the relaxation time of an optically induced, spatially periodic concentration distribution of photochromic molecules. Here, interferences between two coherent plane waves issued from a pulsed high-power laser create a periodic concentration of photoexcited molecules by bringing the photochromic dye molecules in the bright fringes to higher excited states. Due to this concentration distribution, an optical grating is formed because the optical polarizability of the dye molecules has been modified. Following the flash excitation, the concentration distribution will be gradually smeared out by diffusion. The optical grating will relax with a time r characteristic of the translational mass diffusion coefficient D. The probe is a laser beam diffracted by the grating. The scattered electric field is proportional to the amplitude of the concentration distribution. The various scattered electric fields emitted by different parts of the illuminated volume are collected on a phototube so they add up constructively. The output voltage of the phototube will decay with the characteristic time z. This method can be used with photochromic samples or samples doped with comparable size photochromic molecules. The characteristic lifetime of the photoexcited state must be much larger than the time of diffusion of the molecules over the diffusion length equal to the fringe spacing (I-100 #m). This method is suitable for low diffusion coefficients (10-5-10 -ti cm2/s). Its accuracy is of the order of 5%. When the laser excitation energy is converted into heat, a thermal grating is built. It relaxes by thermal diffusion, and the same method can be used to measure the thermal diffusivity of fluids. MISCELLANEOUS PROPERTIES

sin(1/2)(,4 + D ) sin(A/2)

Surface Tension of Fluids The surface tension of fluids is of considerable significance in technical processes involving, for example, boiling, condensation, and multiphase flows. Various experimental methods have been proposed for its determination. The reader will find detailed descriptions of principles and apparatus in Refs. 77 and 78. The capillary method and the method o f the shape o f the sessile drop are referred to as static methods [79, 80], while the method o f the weight or volume o f a drop, the ring or plate detachment method, and the method o f maximal pressure in a gas bubble or drop are quasistatic.

The simplest and most accurate method involves the spontaneous rise of a liquid in capillary tubing. It can be used over a wide temperature range. The mathematical theory is extremely simple. The surface tension is given by o = hgr(pL -- Pv)/2 cos 0

Refractive Index and Dielectric Constant Measurements Refractive Index Determination M e t h o d o f minim u m deviation In this method [52] the sample is conrained in a hollow cell with windows set at an angle to each other, forming a fluid prism of angle A. The refractive index n is deduced from n =

Dielectric Conslant Measurements Parallel-plate or coaxial-cylinder [76] capacitors have been used to measure the dielectric constant e of fluids. The capacitance is measured by a transformer bridge method. Guard rings and guard cylinders allow measurement of the capacitance between ungrounded electrodes of the cell; effects from shielded leads or other capacitances to ground are eliminated by the guard circuit action. Under pressure, the mechanical deformation of the cell has to be taken into account for an accurate determination of the dielectric constant. A 10-2% accuracy on e can be easily obtained and relative variation of E of one part in 106 may be detected.

(24)

where D is the angle of minimum deviation of a monochromatic light beam by the fluid prism. The window faces must be highly polished and parallel to better than 30 seconds of arc. The angle of minimum deviation may be measured with a 10" accuracy, thus determining the refractive index with a 10 -2 % accuracy. Interferometric m e t h o d [74, 75] The cell consists of two identical parts closed at their ends with windows. One part of the cell is filled with the gas, and the other is evacuated. The incident light is split into two beams by an interferometer mirror, one traveling in the filled part and the other in the evacuated part. Interference fringes are seen as the two beams are recombined. The absolute value of the refractive index of the gas is determined by counting the number of fringes that pass a reference point when the cell is evacuated. This method allows very precise measurements of the variation of the index of refraction and consequently of density or composition variation of the fluid.

(25)

where h is the height of rise of the liquid in the tube, r is the radius of the capillary tubing, OL is the density of the liquid, pg is the density of the vapor phase, g is the acceleration due to gravity, and 0 is the contact angle. The accuracy of the measurements is limited mainly to the design of the capillaries and any procedural errors introduced by the operator. Various capillary rise apparatuses have been designed either to meet special experimental conditions [79] or to obtain precise results with a minimum of liquid. The capillary rise method is accepted as the ultimate standard for the determination of liquid surface tension. It generally uses two capillaries of different radii rj and r2, and one measures the vertical distance Ah between the meniscii in the two capillaries: Ah_

2ocosO

(1

g(PL -- PV)

1)

(26)

r2

For liquids that form an appreciable glass-liquid contact angle (0 # 0), the alternative methods such as the maximum bubble pressure of a ga s in the liquid or the weight of a drop are used. CONCLUSION A summary of most of the experimental methods used for the determination of the thermophysical properties of fluids is given in Table 1. This review is far from exhaustive, and the choice of the cited references is certainly arbitrary. The aim

120 R. Tufeu

Table 1. Summary of Experimental Methods Covered in the Text, With Experimental Uncertainties

Measured Property

Experimental Method

Experimental Uncertainties

Equation of state; P, MPa; V, mVkg, T, K

Isochoric method

0.1% for 20
allexperimentaluncertaintieson V

Variable-volume method

0.1% 0.2% 0.2% 0.1% 0.2%

Burnett method Buoyancy force device Vibrating tube desimeter

0.02-0.1% for 80< T< 600 K; low and moderate pressures, generally P < 50 MPa 0.015% for 0 . 1 < P < 1 2 MPa, 273
Sound velocity u, ms-

Interferometric method, tube Interferometric method, sphere Pulse method Echo-overlap method Sing-around method Light scattering (hypersonic)

0.02% for 0.1 < P < 7 MPa, 273.15~
Enthalpy H, J/kg; heat capacity Cp, C~, C,, J/(kg K)

Isothermal calorimeter

0.3-0.7% on H for P=0.1 MPa, 850< T<950 K [23]

Heat flux calorimeter Adiabatic calorimeter (nonflow) Adiabatic calorimeter (flow calorimeter)

0.5% (manuf. spec.) for P=0.1 MPa, 300
Steady-state method, parallel plates

1% liquids, for P = 0.1 MPa, 275 < T< 345 K [40] 1% in critical region [41]

Steady-state method, concentric cylinders

1.5% for 0.1 < P < 100 MPa, 293< T<630 K [42]

Steady-state method, hot wire

I% for 0 . 1 < P < 2 0 MPa, 298.45~
Transient method, hot wire

0.3% for 0.1 < P < I0 MPa, 300< T<425 K [50] 0.5% liquid,for P=0.1 MPa, 2 7 4 < T < 3 1 9 K [51]

Thermal diffusivity k, m2/s

Light scattering

5% author'sestimation;0.02< IT - Tel <50 K, 0.2

Dynamic viscosity ~/, Pas

Capillary flow method

3%, gas, for 0.1
Thermal conductivityk, W/(m K)

Oscillating disk viscometer Oscillating cylinder viscometer Falling body viscometer Rotating cylinder method Torsional quartz viscometer Vibrating wire viscometer Diffusion coefficient D, D~2, m:/s

Loschmidt technique Diaphragm cell Taylor dispersion technique Spin-echo technique Forced Rayleigh scattering method

for 100
0.1%, gas, for 0.1
Refractive index n, dimensionless

Method of minimum deviation (prism) lnterferometric method

10-2%, gas and liquid, 0.02< I T - Tel <50 K, 0.2

Dielectric constant e, dimensionless

Coaxial cylinder capacitor

10-2% for 0.1
10-2% for 0.I < P < 4 5 MPa, T=323.15 K [75]

Thermophysical Properties of Fluids of the present paper was to give some references that could be helpful to novice readers.

NOMENCLATURE

a c

Cp

Cv Cs D, D12

f

sphere radius, wire radius, capillary tubing radius, m velocity of light, m/s specific heat at constant pressure, J/(kg K) specific heat at constant volume, J/(kg K) specific heat at saturation, J/(kg K) diffusion coefficient, m2/s frequency, Hz

g H k M n P q

acceleration due to gravity, m/s 2 enthalpy, J/kg thermal diffusivity, m2/s molar mass, kg/mole refractive index, dimensionless pressure, Pa wave number of light, m -1 O heat flux, W/m2 Re Reynolds number, dimensionless S surface area, m 2 t time, s T temperature, K rc critical temperature, K u sound velocity, m/s y velocity, m/s V specific volume, m3/kg x molecular fraction, dimensionless Z compressibility factor, dimensionless Greek Symbols c~ molecular polarizability, m 3 a e expansion coefficient, K -1 3' gyromagnetic ratio, T -1 s -1 dielectric constant, dimensionless dynamic viscosity, Pa. s 0 angle, radian )~ thermal conductivity, W/(m K) p kinematic viscosity, m2/s p density, kg/m 3 Pc ' critical density, kg/m 3 o surface tension, N/m z characteristic time, s qb angular deflection, rad o~ angular velocity, rad/s

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121

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Received April 15, 1988; revised March 30, 1989.