R. W.
F. T A l T
B. A.
H I L L S
Classz3cation o f the many methods available f o r measuring liquid thermal conductivities, with their limitations and advantages, facilitates the best choice being made in each case
METHODS FOR DETERMINING LIQUID THERMAL CONDUCTIVITIES he thermal conductivity ( k ) of any substance is
Tdefined by the fundamental law of conduction which may be stated differentially as
(dQe/dO) = - k A ( d T / d x ) (11 where dQ, is heat transmitted in time de along a temperature gradient ( d T / d x ) perpendicular to an area A . Thus any method for determining absolute values of k requires the establishment of a known temperature distribution with simultaneous measurement of heat flux. Apparatus for the determination of thermal conductivity may conveniently be divided into two main classes, viz., those in which local values of (dQ,/de) are constant with respect to time (steady-state methods) and those in which local values of (dQ,/dO) vary with respect to time (transient methods). I n the former case, the interpretation of the results is governed by the symmetry of the mathematical model to which Equation 1 is applied, while in the latter cases, interpretation is more dependent upon the phase in which temperature is measured. Hence the following classification is suggested and is used throughout this paper. -Steady-state systems in which isotherms are planar, cylindrical, or spherical. Such methods measure the energy exchange between a heat source and sink, each maintained at a constant temperature. -Transient systems recording the secular change of temperature in the heat source or the liquid. Additional factors to be considered in each category are the thickness of the liquid film and whether or not the determinations are comparative or absolute. Apparatus are generally referred to as “thin film” when the thickness of the liquid sample is such that no appreciable convection currents will arise even after a very long time a t the given temperature and heat flux. I n “thick film” apparatus, on the other hand, convection currents develop soon after the commencement of heating. Thin film apparatus are thus more suited to steady-state methods whereas results from thick film apparatus generally require interpretation upon a transient basis. There are frequently several alternative means for obtaining k from the temperature distribution, each complying with the basic expression describing the:
conducting of heat within the system. These are: extrapolated, using discontinuities in the over-all temperature distribution to locate boundaries ; differential, determining thermal conductivity at a point in the liquid from the temperature distribution in that phase alone ; over-all, assuming the temperatures of the sample boundaries to be those of thermoprobes in the walls. This latter method gives an approximate answer rapidly. I n the following, the practical development of each of these methods is discussed. Steady-State Methods
Planar isotherms. The early investigators realized that the temperature distribution least likely to induce fluid motion was that for which the density increased with depth but remained uniform horizontally. Thus, in their first qualitative observations upon the conduction of heat in liquids, Thomson (125), Nicholson (95), Murray (92), and Rumford (106) set the trend for suppressing convection by passing heat downward from a flat horizontal plate into a tall column of the fluid. Following the acceptance of Biot’s fundamental law of conduction (76), the first determinations of k were made upon similar systems, a cooled lower horizontal plate acting as the heat sink. Variations on this theme were described by Depretz (43), Angstrom ( 6 ) , Paalzov (97), Bottomley (20-22), Chree (35), and R . Weber (732). However, in these early methods, the lack of proper guarding to eliminate side losses and convection together with relatively crude means of measuring temperature and heat flow, undoubtedly caused serious errors. Upon realization of these, the length of the heat path was reduced, the standard apparatus emerging as one which measured the heat flux across a thin film of the liquid sandwiched between the parallel faces of thick copper. disks made from a good conductor-usually All points in each end plate were assumed to be a t the same temperature, i.e., “overall interpretation” was used. T h e temperature differential was maintained by heating one end-plate and cooling the other, the heat flow being calculated from the energy supplied to the first, or that removed from the second, or an average of both. VOL. 5 6
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Another advance in the reduction of errors was the introduction of guard rings, probably first applied by Berget (14) in his work on mercury. T h e thin film method was used in various forms by Guthrie ( 6 0 ) , Beetz (13), H . F. Weber (131), and Lorberg (85). Christiansen (38) compared the thermal conductivities of various liquids with that of air by their use in alternate layers in series to the same heat flux. Copper plates separated each pair of layers. This approach was refined by Henneberg (65),de Heen ( 6 2 ) , Jager (71), Barus (7, S ) , Wachsmith (130), Lees (83), Milner and Chattock (go), etc. By the time of Jakob ( 6 9 ) , parallel plate methods had been developed to the present basic form in which no quantities other than temperature and electrical energy require measurement. Jakob’s apparatus consisted of a 2.5-mms thick liquid layer between two horizontal copper plates, the upper in contact with an electrically heated oil bath and the lower resting on a heat sink held a t a constant temperature. The heat transmitted was estimated from the electrical energy supplied to a coil immersed in the oil bath. Corrections were necessary for convection and radiation losses, although these were reduced by covering the upper part of the apparatus with a vacuum flask. Operating with a temperature difference of 7’ C., Kaye and Higgins (77) refined the apparatus further, largely accounting for erroneous heat losses by effecting a total energy balance to within 5-10yo. Methods similar to this were applied by Martin and Lang (88), Erk and Keller (44), and Kraus (79), while Riedel’s modification ( 103) comprised surrounding the heated plate with two cooling plates, the liquid occupying the intervening spaces. Although strong objections to the “overall” method of interpretation were raised by Chree as early a s 1887, Bates ( 9 ) appears to be the first to have given a detailed description of an apparatus for obtaining thermal conductivity from the temperature distribution in a liquid. His method has been favorably reviewed by many later authors and probably represents the ultimate development of the flat disk arrangement, using conventional means of temperature measurement. b’ithin an annular guard ring, he used flat copper plates 3 in. diameter, the upper plate being electrically heated and the lower a water-cooled calorimeter. T h e steadystate temperature distribution was measured by eight thermocouples stretched across isothermal planes in which they could be moved longitudinally for the detection of convection currents. T h e thermocouples were spring loaded for exact location. JYith a 40’ C. temperature differential between the copper plates, each thermostated to within 1 0 . 2 5 O C., Bates claimed he could obtain point temperatures to within +0.001 ’ C. Interpreting his readings by bsth “extrapolation” and “differential” techniques, he claimed the existence of a film effect at the copper boundaries-quite distinct from that corresponding to the normal convection COefficients. However, Bates partially withdrew this claim in a later paper (70) after he had chromium-plated the cell walls. 30
INDUSTRIAL A N D ENGINEERING CHEMISTRY
I n connection with this possible surface phenomenon, it is interesting to note the later results of Sutherland (722) who reported a similar effect between copper and N-octadecane. Their conclusions cast considerable doubt upon thermal conductivity values obtained using thin films and any interpretation technique other than differential. Powell and Challoner (107) using polished plated surfaces, obtained values of k 2.6 times greater than Sutherland’s. They used film thicknesses of 0.2 mm. and 0.3 mm. compared with Sutherland’s range of 0.1 mm. to 0.69 mm. Further, Ziebland and Patient (138), using plates of 979.1, Cu 3 7 , Ni, could not reproduce Sutherland’s results. Although none of these later investigators appears to have referred to the work of Bates, the possible existence of a surface phenomenon between copper and certain large organic molecules cannot be ignored. Another interesting aspect considered by Bates was the possible variation of thermal conductivity with heat flux. His results indicated that these two quantities were completely independent of each other within the limits afforded by his temperature gradients. This observation is consistent with those of Bridgman (2%?) and Davis (42) upon liquids and that of Wilkes (735) upon refractories. Another approach to the problem of interfacial anomalies was provided in the comprehensive article by Sakiadis and Coates (107) summarizing more detailed accounts (108-11 I ) . They considered location of the surface from the standpoint that some “smooth” surfaces have irregularities of the order of 0.001 inch which may be reduced to 0.0005 inch by plating and polishing. Placing the end plates in direct contact, Sakiadis and Coates obtained an effective surface irregularity of 0.0008 inch but no mention was made as to whether the surfaces had been previously wetted or of the tolerance of their machining. Surface roughness could well amount to 1-2Tc of the plate spacing in thin film methods. Sakiadis and Coates, in their classical determination of the thermal conductivities of 53 common organic liquids, were the first to publish a detailed description of an apparatus in which the sample boundaries could be accurately located by extrapolation of temperature distributions in the neighboring phases. Their method compares liquid conductivities with that of the end plate, the estimated error of &1.5% coming predominantly from the 1 .O% statistically calculated maximum uncertainty in the value of k assumed for steel. They used liquid temperature differentials of 2-6’ C . Eighteen thermocouples placed in the end plates, diametrically staggered, enable the temperature distribution in the steel to be determined, while that of the liquid was obtained by observing the change in its refractive index through the glass retaining walls. Interface and radiation errors were large eliminated by repeating tests for different plate spacings.
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AUTHORS R. W .F . Tait is Projessoi o f Chemical Engineering and B . A . Hdls is Senior Lecturer in Chrmical Engineering at the University o f Adelaide, ilustt alia.
Their results, interpreted by extrapolation, agree essentially with those of other workers similarly determined from thin-film types of apparatus, e.g., Riedel (703-705) and Mason (89). However, these conductivity values are consistently 5% higher than those estimated differentially from thick films using the same temperature readings. Upon the latter basis, the results of Sakiadis and Coates agree with those similarly interpreted by Bates-after correction for his radiation losses. I t is worth noting that this group agrees with the thin film results of Slawecki ( 7 78), who ingeniously obtained the effective cross-sectional area to spacing ratio, J d A / x , from measurements of the electrical capacity. This appears to tip the scales in favor of thick-film methods with differential interpretation, provided convection can be eliminated. Classen and Nedilov (40) first introduced the idea of measuring liquid temperature distributions by refractive index to avoid the disturbance to the heat flow pattern caused by thermocouples. However, this could not be regarded as a precision technique until the developments of Sakiadis and Coates (707) and Frontassjev (49-52), who measured the refraction of a beam of monochromatic light striking a liquid sample of only 2-2.5 ml. perpendicular to the temperature gradient of 1-1.5' C. over-all. T h e method has the disadvantage of requiring the previous calibration of refractive index, but Frontassjev (52) obtained conductivity values for water and toluene over the range 0-70' C. agreeing well with published data. Errors of less than 1% were claimed for this method, which was later improved by Challoner and Powell (30) with the incorporation of guard-rings around a well designed cell in which copper plates differing in temperature by 2-6' C. were used. T h e latest refinement is the thermal comparator described by Clark and Powell (39). Cylindrical isotherms. Reduction of hcat losses from nonisothermal sample boundaries favored by the selection of a system involving radial heat transmission between a heat source and concentric heat sink separated by the liquid. The first apparatus of this type was designed by Schleiermacher (7 72). Improvements by Goldschmidt (55) and Davis (42) led to a form of apparatus consisting of an 0.02 mm. diameter wire stretched axially along a 2-mm. diameter silver tube inserted in a massive copper block. Constant hot-wire temperature was ensured by periodic checks of the wire resistance and the heat generating current. End effects were largely overcome by repeating the experiment with tubes of different length, but otherwise identical. Kardos (74) improved this technique by aligning several tubes of different lengths coaxially. Similar capillary techniques were employed by Timroth and Vargaftik (726), Abas-Zade ( I ) , and Chernajeva (34, while Hutchinson (68) avoided convection errors by striking a different compromise between sample diameter and temperature difference. H e used a tungsten wire maintained 0.5' C. above glass walls of 10 mm. i.d. Reducing the i.d. to 7 mm., Kern and Nostrand
(78) used the same system with the axis horizontal to reduce convection. A comparative method, which avoided the need for exact centering of the hot wire, or knowledge of cell dimensions, was described by Hertz and Fillipov (66). They replaced a liquid of unknown by one of known conductivity in one of two identical cells whose hot wires had previously formed balanced arms of a Wheatstone bridge. T h e circuit was later modified by Fillipov (46) to aid measurement of the unbalance potential. A different form of heat source was introduced by Bridgman (23) in his apparatus primarily designed to investigate the effect of pressure upon fluid conductivity. He measured the thermal resistance between two cylinders, the annular gap containing the liquid being reduced to 0.4 mm. to restrict convection. T h e temperature difference was 0.3' C . Thermosetting and energy supply matters were later improved by Smith (779), Daniloff (47), and Schmidt and Sellschopp ( 7 75). T o reduce the uncertainty of end effects, Riedel (704) (1 948) confined the liquid to the space between a heated copper cylinder, with hemispherical ends, and a complementary hollow block immersed in melting ice. Temperatures were measured by resistance thermometers. Similar apparatus were built by Uhlir (728), Schmidt and Leidenfrost ( 7 74), Petukol- and Kagulin (99),and Challoner (29). A practical means of reducing end effects in cylindrical systems was employed by Woolf and Sibbitt (737) using an axially compensated heat source within the inner of two brass cylinders. T h e 0.065-inch width of the annular gap was later increased by Boggs and Sibbitt (79) to facilitate entry of more viscous liquids. T h e conductivities of liquids in the region of their boiling points have been determined in thin films of condensate for whose deposition the latent heat liberated serves as a heat source. Methods are described by Kallan (72) and Prosad (702). Spherical isotherms. Although a spherical system should eliminate all side or end errors, its application to liquids introduces the need for mechanical support and accurate centering of the inner sphere. Partially on account of construction difficulties such methods have not proved popular. Riedel (105) used the system with thermoprobes inserted near the two parallel spherical surfaces separated by 1 mm. of the liquid. A more recent method, for viscous liquids, is described by Schrock and Starkman (776) who include a detailed analysis of their claim for an accuracy of f1.7%. Transient Methods
General. These methods record temperature as a function of time. Although introduction of the secular response of the detection device is an additional source of error, many other errors may be reduced by interpretation of data, extrapolated back with respect to time to the onset of unsteady conditions. T o render this point distinct, the heat source is restricted to one of low thermal capacity-as manifested by the preponderance of precision transient methods employing VOL. 5 6
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Determination of thermal conductivity thin hot wires. The dimensions of the latter also favor the reduction of radiation losses, and secular extrapolation can largely eliminate the errors caused by convection, calorimetry, and the temperature dependence of thermal conductivity. The optical temperature measurements described previously may be applied to transient systems with equal advantage. Seldom estimated below 0.57,, the dominating error in most “hot wire” determinations of liquid thermal conductivities is that caused by the axial flow of heat toward the filament supports. The latter are cooler by virtue of their lower electrical resistance. By comparison with hot wire techniques, other transient methods warrant no more than a mention of those gaining recognition at their time of publication. These include “flat plate” apparatus from Lundquist (87) to Soonawala ( 720), plus the cylinder versions of Winkelmann (136)and Weber (737), the calorimetric methods of Graetz (56) and Callendar (26),the impressed velocity method of Settleton (93, 94), and that of Shiba (777) using a piezometer. Hot wire methods. The first transient hot wire method was probably that described by Stalhane and Pyk (727) for solids, although it later became standard equipment for liquids. Prior to this period, unsteady state techniques had been avoided on account of the mathematical complexity of interpreting results. However, with the derivation of the equation
Q
= (4dT/log
0)
+ constant
(2)
(Q refers to unit length) by Eucken and Englert (45), k could be determined from the secular variation of the temperature at a point receiving heat radially from a hot wire. The integration constant and the errors from any interfacial resistances could be eliminated by graphical interpretation. Intent upon reducing convection, Pfriem (700) described a method in which the temperature at a selected point was recorded. End effects were eliminated by repeating the experiment with different lengths of wire. In an improved version, Weishaupt (134)measured the change in temperature of the wire from its recorded resistance in a Wheatstone bridge circuit. This permitted the use of low temperature gradients, ideally suited to conditions where convection is easily induced, i.e., for liquids of low viscosity and high thermal expansion. Dependent upon the resistivity of the fluid, thermal conductivity errors approaching + 1.6% were claimed. Whereas Weishaupt essentially measured temperatures averaged over the length of the wire. Fischer (47) preferred to use a thermocouple. Following their theoretical analysis of the relevant transient system, van der Held and van Drunen (63) described the first really practical version of this method. Using a 0.2-mm. diameter manganese wire and 0.1-mm. diameter thermocouple inserted in a thin glass tube immersed in the liquid, they avoided convection errors by completing 32
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their measurements within 10-1 5 seconds of imposing transient conditions. They claimed an accuracy of f2% after making corrections for end effects. Further theoretical considerations were presented by Blackwell (77) and applied recently by Panaitescu and Pauker (98). Reverting to the principle of Schleiermacher’s classical method for gases, Gillam et a l . (54)measured the change in resistance of a 0.1-mm. diameter wire mounted axially in a cylinder maintained to within =tO.005’C. Adopting an interpretation technique which minimized end effects, the method gave a reproducibility of f0.30/, for liquids less prone to develop convection currents. To avoid time lags, Allen (5) recorded the wire resistance upon an oscilloscope in a circuit designed to maintain a constant heating current. However, the many terms appearing in the equation of interpretation reduced the accuracy to his conservative estimate of j=570. With a similar recording device, and quoted error? Hill (67) was able to obtain the thermal conductivity of a liquid surrounding his 0.1-mm. diameter thermocouple heating element subjected to a high-frequency rectangular pulse. Readings were completed within one second. The determination of liquid thermal conductivities from measurements upon the heat source alone was continued by Grassman and Straumann (57) who claimed an accuracy of *l%. Their method automatically recorded the resistance of a hot platinum wire relative to a reference wire in the adjacent arm of a \Vheatstone bridge circuit. Turnbull (727) described a simple, well-designed probe in which the two end effects were almost eliminated by their inclusion in opposing arms of the bridge. This was achieved by taking a potential tapping 1 cm. from one of the supported ends of his 10 cm. x 0.05 mm. diameter hot platinum wire. Although requiring initial temperature calibration, the probe is convenient for use in almost any system, and the accuracy claim of A3yo for inorganic melts over the range 0-400’ C. seems a very conservative figure. Reproducibility was better than 1%. The application of optical means of temperature measurement to transient hot wire methods was described by Lamm (87) using a modified Schlieren shadow photo technique. Commensurate with its steady-state applications, ready detection of any gas bubbles or convection currents was provided. Lamm’s method has been further developed by Bryngdahl (24) but is restricted to transparent liquids. Determinations at High Temperatures
InlTestigators into the thermal conductivities of fluids at high temperatures have tended to select steady-state methods in which the greater potential side heat losses were avoided by surrounding the liquid container with a medium maintained isothermal to the energy source or
usually assumes the validity of Fourier's law sink. This has proved conducive to the use of cylindrical systems with radial heat transmission, as illustrated by Vargaftik (129) for a heat transfer salt melt a t 200-300' C., by Bloom (18) for alkali nitrates at 310-340' C., by Burton (25) for Dowtherm, and by Kroger and Elighausen (80) for melting glass. However, Lucks and Deem (86) preferred a simple flat plate method for their investigations of sodium hydroxide melts a t 250-450' C., using sand as the calibration medium. T h e hot wire method has been applied by Cecil (28) in their determinations of the thermal conductivities of several high molecular weight organic compounds in the region 25-200' C. Turnbull applied his method to eutectic nitrate mixtures in the region of 350' C., the simplicity and remote control of the probe offering little hindrance to the careful thermostatting needed. Methods for liquids Under Shear
Although no direct reference could be found to the determination of the thermal conductivity of liquids under shear, methods were occasionally mentioned in investigations into the heat transfer properties of fluids in turbulent motion. While most workers have been concerned with axial flow, annular systems subject to Couette flow have proved more convenient for the establishment of steady-state conditions. Moreover localized shear rates were easier to estimate and adjust. Methods describing measurements of the transfer of heat between rotating coaxial cylinders have been described by Schlichting (113), Gazley (53) and Kaye and Elgar (75) for use with air. However they were mostly concerned with the more turbulent flow regions as depicted in the classical work of Taylor (124). The apparatus developed by Tachibama (723) used a n electrically heated inner rotating cylinder, with guard rings, for investigation of the heat transfer properties of air, spindle oil, and mobile oil. For speeds of up to 1000 r.p.m. he used five annulus widths with temperature differentials of 20-30" C. The outer cylinder was watercooled. The speed at which the system will diverge from purely laminate flow, and hence radical conduction, could be forrcnst from the Taylor number. Analysis of Errors
Whatever method is adopted, the final determination of a n absolute value for liquid thermal conductivity must invoke application of the fundamental law of conduction or one of its integrated forms. Hence the common sources of error will be discussed in connection with the relevant term in Equation 1. Heat flux (dQc/dO) The greatest difficulty in most methods is the prevention of heat transfer by convection for which error it is virtually impossible to calculate a worthwhile correction. Its suppression is favored by using thin liquid layers, small temperature gradients, and planar horizontal isotherms whose temperature increases with height.
T h e last indicates a slight advantage for parallel plate methods. Convection is probably best detected by optical means although, in many transient methods, it may be sufficiently delayed to enable its onset to be recognized from a discontinuity in temperature readings. Mechanical means of detection, such as those used by Bates, are much more cumbersome and isotherm scanning by thermocouples may tend to induce fluid motion. However a prediction of the tendency for buoyancy forces to overcome inertial and viscous forces may be obtained from the Rayleigh number, or product of the Grashof ( N G ~ and ) Prandl (Npr) numbers. T h e work of Beckmann (12) and Mull and Reiher (91)) with further interpretation by Jakob (70), indicates that convection errors should not exceed 2% if (Np,) (NGr) > 1000. However, this figure might well be considerably reduced for thixotropic liquids. Corrections for radiation can usually be applied, but the uncertainty definitely favors a low temperature gradient and a small heat source, i.e., a hot wire with a ' as opposed to a thin correction factor of the order of 0.1% film apparatus in which it may amount to 5%, However, many practical techniques may be employed to account for radiation such as preliminary runs in vacuo, adjustment of film thickness, etc. Another serious source of error is the loss of heat from the sample to the surroundings, thus distorting isotherms from the mathematical pattern assumed for the integration of Equation l , In this respect cylindrical are preferable to flat plate methods while constructional difficulties have rendered spherical systems unpopular. Such errors in steady-state methods may be minimized using guard rings,.or estimated from a total heat balance. T h e errors caused by axial flow in transient hot wire techniques can be very neatly eliminated by taking potential tappings away from the ends. However, in all systems with cylindrical isotherms, the greater the 1ength:diameter ratio the lower the error. Exact solutions, from which end effects may be estimated, are given by Carslaw and Jaeger (27) for the principal cases encountered. Having considered its partial dissipation and contributing mechanisms, the total heat transmitted may be measured as the energy supplied to the heat source, or that received by the heat sink, or both. T h e former is most simply estimated by using an electric current, while water-calorimetry is normally employed for the latter. While both coiled heating elements and thin elements suffer from end effects, uninsulated wires are only permissible in liquids of electrical conductivity some 1000 fold lower. Since thermal conductivity values for opaque solids must be free from convection and internal radiation errors, their comparison with fluids is often performed using solid and fluid samples in series to the same heat flux. Measurement of Q , is then unnecessary. This VOL. 5 6
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comparison technique \vas well employed by Sakiadis and Coates, but absolute liquid conductivities cannot be obtained to an accuracy within about +0.5y0of that of the reference solid. Area of cross-section (A). Little error is normally introduced in the term ( A ) , or its counterparts in annular or spherical conductors, provided flat plates are parallel and good mirror images of each other, or cylindrical systems are exactly co-axial. Such errors may be avoided by comparison techniques using liquid-liquid replacement in the same sample cell. Visual observation of the liquid is an advantage in ensuring the complete absence of gas bubbles. Temperature gradient (dT ’dx). Possible interfacial resistances, together with the inevitable roughness of the source and sink walls, certainly favors the use of film thicknesses giving thermal resistances large relative to these uncertainties. Although strongly contested, the possible existence of other surface phenomena suggests the avoidance of unplated copper in contact with thin liquid films. Tj\‘hen fluids are quite good conductors, disregard for temperature gradients within the end plates may introduce a n appreciable error. Thus the overall method of interpretation should be avoided for all but the most approximate determinations. Most investigators recording the temperature distribution within a phase seldom mention the disturbance to the heat flow pattern caused by thermoprobes. However, the diametric staggering of such probes reduces the total error to almost that of a n isolated one. The popular use of thermocouples, with junctions down to 0.1-mm. diameter, thus demands a film thickness of 1 cm. for a 1% maximum displacement in the direction of the temperature gradient. Differential or extrapolated interpretation of results from within the liquid sample thus favors the use of the widest fluid film permitted by convection. Disturbance errors are avoided by any optical method, or by techniques recording hot wire resistance only. T h e error in the temperature gradient is reduced by increased differential between the heat source and sink, with thermocouples and optical interference techniques claiming a constant yet somewhat optimistic accuracy of ~ 0 . 0 0 1 ’C. Allowing h00.0020C. the minimum temperature drop, measurable to a probable error of h 1% by two thermocouples, would be 2.8’ C. \$’hereas initial changes in source temperatures can be taken into account in transient methods, the relatively slow response of most steady-state systems permits thermostating tolerances of up to =t00.050C. T h e most sophisticated form of temperature control quoted in descriptions of these methods uses a thermistor probe to complete a circuit supplying about 10% of the power necessary to maintain the bath a few degrees below the temperature required. Thermal conductivity (k). IVhereas k appears independent of heat flux for all reasonable experimental conditions, it can vary with temperature and molecular orientation. The latter anisotropy can probably be induced under shear. Unless a mathematical relationship 34
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
is assumed initially, or very small differentials are used, the temperature dependence of thermal conductivity can only be determined by the differential interpretation of results. Reproducibility
Reading descriptions of the foregoing methods, one gains the impression that quoted accuracies have failed to keep pace with the vast improvements in technique over the years. This can be largely attributed to the more realistic consideration of sources of error including many hitherto ignored or dismissed on account of the difficulty of their estimation. An illustration of the unjustified optimism recorded for many acclaimed methods is provided by considering quoted thermal conductivity values for certain simple liquids. Seventeen values for water published by recognized investigators since 1930 averaged 1.444 X cal. cm. by Nukijama sec. ’ C., and ranged from 1.413 X and Yoshijawa ( 9 6 )to 1.490 X lo-’ by van der Held (64), i.e., deviations varying from -2.1% to +3.2%. Similarly, nine of the best values for glycerol recorded since C..and 1928 averaged 6.891 X l o u 4 cal.’cm. sec. ranged from 6.718 X 10+ by Hutchinson (68)to 7.028 X l o p 4 by Mason (89) and 7.325 X 10-4 by Sakiadis and Coates (707), i.e., deviations from -2.5Vc to 7.7%. Discussion
I n selecting the method to be used for determination of the thermal conductivity of a given liquid, one must answer certain questions, e.g. (a)-what degree of accuracy is necessary? (b)-are absolute values required, or will comparative ones suffice? (c)-is time required for the determinations an important factor? (d)-should optical methods of temperature measurement be used? (e)-is cost an important factor? (f)-what should be the dimensions of the apparatus? Question (a) is the most important. If a high degree of accuracy is essential, then it may not be possible to meet some of the other requirements. O n the other hand, if probable errors up to =t3% are acceptable, then one of the relatively low cost steady state methods may be the best solution. However, this would require that the answer to (c) be “Vo.” Where a high degree of accuracy is necessary and the liquid is transparent, the answer to question (d) will be “Yes.” I n using optical methods, radiation corrections are important and must be carefully applied. The answer to question (f) will always require a compromise. Accurate results require complete suppression of convection currents. This in turn requires, for steady state methods, small temperature differences and/or small clearances between heat source and heat sink, or, for transient methods, measurements over brief time intervals so that the required results can be ob-
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